Roman Catholics claim that other Churches do not have Apostolic Succesion because the keys to heaven was given to the first Pope – The Apostle Peter. Peter is the rock on whom the Church was built and thus if you are not part of the Roman Catholic Church and under the authority of the Pope, you are not in the One True Church. Roman Catholics cite Matthew 16:18-19 as proof for papal authority and their doctrine of the One True Church, along with the claim that only Priest can forgive sins. However, if we backtrack and read from verses 13 we read that Peter confeses to Christ that he is the son of God.  We will see that for this reason was Peter named, and not because he was on whom the Church was buiilt, but on his faith:

Matthew 16:13-19

13 When Jesus came into the region of Caesarea Philippi, He asked His disciples, saying, “Who do men say that I, the Son of Man, am?” 14 So they said, “Some say John the Baptist, some Elijah, and others Jeremiah or one of the prophets.” 15 He said to them, “But who do you say that I am?” 16 Simon Peter answered and said, “You are the Christ, the Son of the living God.”17 Jesus answered and said to him, “Blessed are you, Simon Bar-Jonah, for flesh and blood has not revealed this to you, but My Father who is in heaven. 18 And I also say to you that you are Peter, and on this rock I will build My church, and the gates of Hades shall not [g]prevail against it. 19 And I will give you the keys of the kingdom of heaven, and whatever you bind on earth [h]will be bound in heaven, and whatever you loose on earth will be loosed in heaven.”20 Then He commanded His disciples that they should tell no one that He was Jesus the Christ. – NKJV

Comparing the early Church Fathers quotes on the above Scripture, we can see that they believed it was Peter’s faith, not on his person, that the Church was built:

John Chrysostom:

“”Upon this rock,” not upon Peter. For He built His Church not upon man, but upon the faith of Peter. But what was his faith? “Thou art the Christ, the Son of the living God.” – John Chrysostom

Hilary:

“To Peter the Father revealed that he should say, “Thou art the Son of the living God.” Therefore the building of the Church is upon this rock of confession; this faith is the foundation of the Church.” – Hilary

If you believe Jesus is the son of God, you are part of the Church.

The gates of Hades/Hell – death – will not prevail against anyone who has this same faith:

Origen:

And after this let us see in what sense it is said to Peter, and to every Peter, “I will give unto thee the keys of the kingdom of heaven. And, in the first place, I think that the saying, “I will give unto thee the keys of the kingdom of heaven,” is spoken in consistency with the words, “The gates of Hades shall not prevail against it.” For he is worthy to receive from the same Word the keys of the kingdom of heaven, who is fortified against the gates of Hades so that they do not prevail against him, receiving, as it were, for a prize, the keys of the kingdom of heaven, because the gates of Hades had no power against him, that he might open for himself the gates that were closed to those who had been conquered by the gates of Hades. And he enters in, as a temperate man, through an opened gate—the gate of temperance—by the key which opens temperance; and, as a righteous man, by another gate—the gate of righteousness—which is opened by the key of righteousness; and so with the rest of the virtues.  – Origen, Commentary on Matthew, XII

The Promise Given to Peter Not Restricted to Him, But Applicable to All Disciples Like Him.

The Church was built upon the other Apostles, not just Peter. And the keys were given equally to them and to anyone who holds to the same common faith and we are all a Peter in this way:

But if you suppose that upon that one Peter only the whole church is built by God, what would you say about John the son of thunder or each one of the Apostles? Shall we otherwise dare to say, that against Peter in particular the gates of Hades shall not prevail, but that they shall prevail against the other Apostles and the perfect? Does not the saying previously made, “The gates of Hades shall not prevail against it,”  hold in regard to all and in the case of each of them? And also the saying, “Upon this rock I will build My church”? Are the keys of the kingdom of heaven given by the Lord to Peter only, and will no other of the blessed receive them? But if this promise, “I will give unto thee the keys of the kingdom of heaven,”  be common to the others, how shall not all the things previously spoken of, and the things which are subjoined as having been addressed to Peter, be common to them? For in this place these words seem to be addressed as to Peter only, “Whatsoever thou shalt bind on earth shall be bound in heaven,” but in the Gospel of John the Saviour having given the Holy Spirit unto the disciples by breathing upon them said, “Receive ye the Holy Spirit,”  Many then will say to the Saviour, “Thou art the Christ, the Son of the living God;” but not all who say this will say it to Him, as not at all having learned it by the revelation of flesh and blood but by the Father in heaven Himself taking away the veil that lay upon their heart, in order that after this “with unveiled face reflecting as a mirror the glory of the Lord”  they may speak through the Spirit of God saying concerning Him, “Lord Jesus,” and to Him, “Thou art the Christ, the Son of the living God.” And if any one says this to Him, not by flesh and blood revealing it unto Him but through the Father in heaven, he will obtain the things that were spoken according to the letter of the Gospel to that Peter, but, as the spirit of the Gospel teaches, to every one who becomes such as that Peter was. For all bear the surname of “rock” who are the imitators of Christ, – ibid

And if we too have said like Peter, “Thou art the Christ, the Son of the living God,” not as if flesh and blood had revealed it unto us, but by light from the Father in heaven having shone in our heart, we become a Peter, and to us there might be said by the Word, “Thou art Peter,” For a rock is every disciple of Christ of whom those drank who drank of the spiritual rock which followed them, and upon every such rock is built every word of the church, and the polity in accordance with it; for in each of the perfect, who have the combination of words and deeds and thoughts which fill up the blessedness, is the church built by God.-ibid

Peter was not first among the Apostles in authority. He was the first to make the confession of Christ as the son of God. Likewise this had nothing to do with the power of forgiving sins:

Tertuallian:

Peter was the first of all to be endued with the Spirit, and, after making preface touching the calling of the nations, to say, “And now why are ye tempting the Lord, concerning the imposition upon the brethren of a yoke which neither we nor our fathers were able to support? But however, through the grace of Jesus we believe that we shall be saved in the same way as they.” This sentence both “loosed” those parts of the law which were abandoned, and “bound” those which were reserved. Hence the power of loosing and of binding committed to Peter had nothing to do with the capital sins of believers; and if the Lord had given him a precept that he must grant pardon to a brother sinning against him even “seventy times sevenfold,” of course He would have commanded him to “bind”–that is, to “retain”–nothing subsequently, unless perchance such (sins) as one may have committed against the Lord, not against a brother. For the forgiveness of (sins) committed in the case of a man is a prejudgment against the remission of sins against God.

What, now, (has this to do) with the Church, and) your (church), indeed, Psychic? For, in accordance with the person of Peter,it is to spiritual  men that this power will correspondently appertain, either to an apostle or else to a prophet. For the very Church itself is, properly and principally, the Spirit Himself, in whom is the Trinity of the One Divinity–Father, Son. and Holy Spirit. (The Spirit) combines that Church which the Lord has made to consist in “three.” And thus, from that time forward, every number (of persons) who may have combined together into this faith is accounted “a Church,” from the Author and Consecrator (of the Church). And accordingly “the Church,” it is true, will forgive sins: but (it will be) the Church of the Spirit, by means of a spiritual man; not the Church which consists of a number of bishops. For the right and arbitrament is the Lord’s, not the servant’s; God’s Himself, not the priest’s. – Tertuallian, On Modesty

“They then (the apostles) in like manner founded Churches in every city, from which all the other Churches, one after another, derived the tradition of the faith and the seeds of doctrine, and are every day deriving them, that they may become Churches. Indeed it is on this account only that they will be able to account themselves apostolic, as being the offspring of apostolic Churches. . . . Therefore the Churches, although they are so many and so great, comprise but the one primitive Church founded by the apostles from which they all spring. In this way all are primitive, and all are apostolic, whilst they are all proved to be one, in unity, by their salutation of peace (communicatio pacis), and title of brotherhood, and bond of hospitality (contesseratio hospitalitatis)—rights which no other rule directs than the one tradition of the self-same mystery.” – IBID

Hilary of Poitiers:

A belief that the Son of God is Son in name only, and not in nature, is not the faith of the Gospels and of the Apostles…whence I ask, was it that the blessed Simon Bar–Jona confessed to Him, Thou art the Christ, the Son of the living God?…And this is the rock of confession whereon the Church is built…that Christ must be not only named, but believed, the Son of God. This faith is that which is the foundation of the Church; through this faith the gates of hell cannot prevail against her. This is the faith which has the keys of the kingdom of heaven. Whatsoever this faith shall have loosed or bound on earth shall be loosed or bound in heaven…The very reason why he is blessed is that he confessed the Son of God. This is the Father’s revelation, this the foundation of the Church, this the assurance of her permanence. Hence has she the keys of the kingdom of heaven, hence judgment in heaven and judgment on earth….Thus our one immovable foundation, our one blissful rock of faith, is the confession from Peter’s mouth, Thou art the Son of the living God

Augustine:

“In a passage in this book, I said about the Apostle Peter: ‘On him as on a rock the Church was built’…But I know that very frequently at a later time, I so explained what the Lord said: ‘Thou art Peter, and upon this rock I will build my Church,’ that it be understood as built upon Him whom Peter confessed saying: ‘Thou art the Christ, the Son of the living God,’ and so Peter, called after this rock, represented the person of the Church which is built upon this rock, and has received ‘the keys of the kingdom of heaven.’ For, ‘Thou art Peter’ and not ‘Thou art the rock’ was said to him. But ‘the rock was Christ,’ in confessing whom, as also the whole Church confesses, Simon was called Peter. But let the reader decide which of these two opinions is the more probable.” – Augustine

Remember, in this man Peter, the rock. He’s the one, you see, who on being questioned by the Lord about who the disciples said he was, replied, ‘You are the Christ, the Son of the living God.’ On hearing this, Jesus said to him, ‘Blessed are you, Simon Bar Jona, because flesh and blood did not reveal it to you, but my Father who is in heaven. And I tell you’…’You are Peter, Rocky, and on this rock I shall build my Church, and the gates of the underworld will not conquer her. To you shall I give the keys of the kingdom. Whatever you bind on earth shall also be bound in heaven; whatever you loose on earth shall also be loosed in heaven’ (Mt 16:15-19). In Peter, Rocky, we see our attention drawn to the rock. Now the apostle Paul says about the former people, ‘They drank from the spiritual rock that was following them; but the rock was Christ’ (1 Cor 10:4). So this disciple is called Rocky from the rock, like Christian from Christ. Why have I wanted to make this little introduction? In order to suggest to you that in Peter the Church is to be recognized. Christ, you see, built his Church not on a man but on Peter’s confession. What is Peter’s confession? ‘You are the Christ, the Son of the living God.’ There’s the rock for you, there’s the foundation, there’s where the Church has been built, which the gates of the underworld cannot conquer. – Augustine

“But whom say ye that I am? Peter answered, ‘Thou art the Christ, The Son of the living God.’ One for many gave the answer, Unity in many. Then said the Lord to him, ‘Blessed art thou, Simon Barjonas: for flesh and blood hath not revealed it unto thee, but My Father which is in heaven.’ Then He added, ‘and I say unto thee.’ As if He had said, ‘Because thou hast said unto Me, “Thou art the Christ the Son of the living God;” I also say unto thee, “Thou art Peter.” ’ For before he was called Simon. Now this name of Peter was given him by the Lord, and in a figure, that he should signify the Church. For seeing that Christ is the rock (Petra), Peter is the Christian people. For the rock (Petra) is the original name. Therefore Peter is so called from the rock; not the rock from Peter; as Christ is not called Christ from the Christian, but the Christian from Christ. ‘Therefore,’ he saith, ‘Thou art Peter; and upon this Rock’ which Thou hast confessed, upon this rock which Thou hast acknowledged, saying, ‘Thou art the Christ, the Son of the living God, will I build My Church;’ that is upon Myself, the Son of the living God, ‘will I build My Church.’ I will build thee upon Myself, not Myself upon Thee. For men who wished to be built upon men, said, ‘I am of Paul; and I of Apollos; and I of Cephas,’ who is Peter. But others who did not wish to built upon Peter, but upon the Rock, said, ‘But I am of Christ.’ And when the Apostle Paul ascertained that he was chosen, and Christ despised, he said, ‘Is Christ divided? was Paul crucified for you? or were ye baptized in the name of Paul?’ And, as not in the name of Paul, so neither in the name of Peter; but in the name of Christ: that Peter might be built upon the Rock, not the Rock upon Peter. This same Peter therefore who had been by the Rock pronounced ‘blessed,’ bearing the figure of the Church.” –  Augustine

Ambrose:

He, then, who before was silent, to teach us that we ought not to repeat the words of the impious, this one, I say, when he heard: ‘But who do you say I am,’ immediately, not unmindful of his station, exercised his primacy, that is, the primacy of confession, not of honor; the primacy of belief, not of rank…This, then, is Peter who has replied for the rest of the Apostles; rather, before the rest of men. And so he is called the foundation, because he knows how to preserve not only his own but the common foundation. Christ agreed with him; the Father revealed it to him. For he who speaks of the true generation of the Father, received it from the Father, did not receive it from the flesh. Faith, then, is the foundation of the Church, for it was not said of Peter’s flesh, but of his faith, that ‘the gates of hell shall not prevail against it.’ But his confession of faith conquered hell. And this confession did not shut out one heresy, for, since the Church like a good ship is often buffeted by many waves, the foundation of the Church should prevail against all heresies. The day will fail me sooner than the names of heretics and the different sects, yet against all is this general faith-that Christ is the Son of God, and eternal from the Father, and born of the Virgin Mary. –IBID

Athanasius:

I know moreover that not only this thing saddens you, but also the fact that while others have obtained the churches by violence, you are meanwhile cast out from your places. For they hold the places, but you the Apostolic Faith. They are, it is true, in the places, but outside of the true Faith; while you are outside the places indeed, but the Faith, within you…But ye are blessed, who by faith are in the Church, dwell upon the foundations of the faith, and have full satisfaction, even the highest degree of faith which remains among you unshaken. For it has come down to you from Apostolic tradition, and frequently has accursed envy wished to unsettle it, but has not been able. On the contrary, they have rather been cut off from their attempts to do so. For thus it is written, ‘Thou art the Son of the Living God,’ Peter confessing it by revelation of the Father, and being told, ‘Blessed art thou Simon Barjona, for flesh and blood did not reveal it to thee, but My Father Who is in heaven,’ and the rest. No one therefore will ever prevail against your Faith, most beloved brethren.- Athanasius

Eusebius:

“I, therefore, brethren, who have lived sixty-five years in the Lord, and have met with the brethren throughout the world, and have gone through every Holy Scripture, am not affrighted by terrifying words. For those greater than I have said, we ought to obey God rather than man. ” -Eusebius, quoting Letter to Victor by Polycrates

Jerome:

The one foundation which the apostolic architect laid is our Lord Jesus Christ. Upon this stable and firm foundation, which has itself been laid on solid ground, the Church of Christ is built…For the Church was founded upon a rock…upon this rock the Lord established his Church; and the apostle Peter received his name from this rock.

John of Damascus:

This rock was Christ, the incarnate Word of God, the Lord, for Paul clearly teaches us: ‘The rock was Christ’

According to John Chrysostom, the Church – body of Christ – was given the keys to heaven, not Peter:

For the Son of thunder, the beloved of Christ, the pillar of the Churches throughout the world, who holds the keys of heaven, who drank the cup of Christ, and was baptized with His baptism, who lay upon his Master’s bosom, with much confidence, this man now comes forward to us now – IBID

Roman Catholic Quotes of the Early Church Fathers are often altered to not reflect what was actually being conveyed. Here is one example from Cyprian:

CYPRIAN OF CARTHAGE:

This is the common Roman Catholic Version of the following quote:

“The Lord says to Peter: ‘I say to you,’ he says, ‘that you are Peter, and upon this rock I will build my Church, and the gates of hell will not overcome it. … ’ [Matt. 16:18]. On him [Peter] he builds the Church, and to him he gives the command to feed the sheep [John 21:17], and although he assigns a like power to all the apostles, yet he founded a single chair [cathedra], and he established by his own authority a source and an intrinsic reason for that unity. . . . If someone [today] does not hold fast to this unity of Peter, can he imagine that he still holds the faith? If he [should] desert the chair of Peter upon whom the Church was built, can he still be confident that he is in the Church?” (The Unity of the Catholic Church 4; first edition [A.D. 251]).

This is the Quote in Roberts/Donaldson:

If any one consider and examine these things, there is no need for lengthened discussion and arguments. There is easy proof for faith in a short summary of the truth. The Lord speaks to Peter, saying, I say unto you, that you are Peter; and upon this rock I will build my Church, and the gates of hell shall not prevail against it. And I will give unto you the keys of the kingdom of heaven; and whatsoever you shall bind on earth shall be bound also in heaven, and whatsoever you shall loose on earth shall be loosed in heaven. And again to the same He says, after His resurrection, Feed my sheep. And although to all the apostles, after His resurrection, He gives an equal power, and says, As the Father has sent me, even so send I you: Receive the Holy Ghost: Whose so ever sins you remit, they shall be remitted unto him; and whose soever sins you retain, they shall be retained; (John 20:21) yet, that He might set forth unity, He arranged by His authority the origin of that unity, as beginning from one. Assuredly the rest of the apostles were also the same as was Peter, endowed with a like partnership both of honour and power; but the beginning proceeds from unity.

Here is another quote from Cyprian that is altered by Roman Catholics:

“There is one God and one Christ, and one Church, and one chair founded on Peter by the word of the Lord. It is not possible to set up another altar or for there to be another priesthood besides that one altar and that one priesthood. Whoever has gathered elsewhere is scattering” -Cyprian

Here is the same quote unaltered:

There is one God, and Christ is one, and there is one Church, and one chair founded upon the rock by the word of the Lord. Another altar cannot be constituted nor a new priesthood be made, except the one altar and the one priesthood. Whosoever gathereth elsewhere, scattereth. Whatsoever is appointed by human madness, so that the divine disposition is violated, is adulterous, is impious, is sacrilegious. Depart far from the contagion of men of this kind. and flee from their words, avoiding them as a cancer and a plague, as the Lord warns you and says, “They are blind leaders of the blind. But if the blind lead the blind, they shall both fall into the ditch.” -IBID

All the Apostles were equal:

Clement of Alexandria:

To James the Just, and John and Peter, the Lord after His resurrection imparted knowledge.  These imparted it to the rest of the apostles, and the rest of the apostles to the Seventy, of whom Barnabas was one. – Clement of Alexandria, Hypotyposes, Book 6 quoted in Eusebius, Church History

James, not Peter, lead the council at Jerusalem. John Chrysostom pointed this fact out citing that James was the Bishop, not Peter:

John Chrysostom:

This (James) was bishop, as they say, and therefore he speaks last..There was no arrogance in the Church. After Peter Paul speaks, and none silences him: James waits patiently; not starts up (for the next word). No word speaks John here, no word the other Apostles, but held their peace, for James was invested with the chief rule, and think it no hardship. So clean was their soul from love of glory. Peter indeed spoke more strongly, but James here more mildly: for thus it behooves one in high authority, to leave what is unpleasant for others to say, while he himself appears in the milder part -John Chrysostom

John Calvin:

Let Peter have the preeminence, still there is a great difference between the honour of rank and the possession of power. We see that the Apostles usually left it to Peter to address the meeting, and in some measure take precedence in relating, exhorting, admonishing, but we nowhere read anything at all of power. –  John Calvin, Institutes  CHAPTER 6. OF THE PRIMACY OF THE ROMISH SEE.

Maintaining the teachings of the Apostles is what shows the Churches to be in “succession”, not the chair of Peter:

Irenaeus:

Wherefore it is incumbent to obey the presbyters who are in the Church,- those who, as I have shown, possess the succession from the apostles; those who, together with the succession of the episcopate, have received the certain gift of truth, according to the good pleasure of the Father. But [it is also incumbent] to hold in suspicion others who depart from the primitive succession, and assemble themselves together in any place whatsoever, [looking upon them] either as heretics of perverse minds, or as schematics puffed up and self-pleasing, or again as hypocrites, acting thus for the sake of lucre and vainglory. For all these have fallen from the truth. And the heretics, indeed, who bring strange fire to the altar of God – namely, strange doctrines – shall be burned up by the fire from heaven, as were Nadab and Abiud. -Irenaeus

Clement of Alexandria:

“Luminous already, and like the sun shining in the exercise of beneficence, he speeds by righteous knowledge through the love of God to the sacred abode, like as the apostles. . . . Those then also, who have exercised themselves in the Lord’s commandments, and lived perfectly and gnostically according to the Gospel may be now enrolled in the chosen body of the apostles. Such an one is in reality an elder of the Church, and a true deacon of the will of God if he do and teach what is the Lord’s; not as being chosen by men, nor regarded as righteous because a presbyter, but enrolled in the eldership because righteous. And although here upon earth he be not honoured with the chief seat, he will sit down on the four-and-twenty thrones, judging the people, as St. John says in the Apocalypse. For in truth the covenant of salvation, reaching down to us from the foundation of the world, through different generations and times, is one, though conceived as different in respect of gifts.”

Tertullian

“None of the Heretics Claim Succession from the Apostles. Because Their Faith is that Which the Apostles Taught and Handed Down.

The Early Church Fathers actually wrote against the idea of a Pope:

Gregory the Dialogist:

“I say it without the least hesitation, whoever calls himself the universal bishop, or desires this title, is by his pride the precursor of the anti-Christ because he thus attempts to raise himself above the others. The error into which he falls springs from pride equal to that of the anti-Christ. For as that wicked one wished to be regardedas exalted above other men, like a god, so likewise whosoever would call himself sole bishop exalteth himself above others”

“It cannot be denied that if any one bishop be called universal, all the Church crumbles if that universal one fall”- ibid

Cyprian:

“For neither does any of us set himself up as a bishop of bishops, nor by tyrannical terror does any compel his colleague to the necessity of obedience; since every bishop, according to the allowance of his liberty and power, has his own proper right of judgment, and can no more be judged by another than he himself can judge another”

Jerome:

“Wherever a bishop may be whether at Rome or at Eugubium, at Constantinople or at Rhegium, at Alexandria or at Thanis, he is of the same worth…for all of them are the successors of the apostles.”

Whatsoever is appointed by human madness, so that the divine disposition is violated, is adulterous, is impious, is sacrilegious. Depart far from the contagion of men of this kind. And flee from their words, avoiding them as a cancer and a plague, as the Lord warns you and says, “They are blind leaders of the blind. But if the blind lead the blind, they shall both fall into the ditch.” – IBID

Recall the earlier quote from Hilary of Poitiers where he stated, “A belief that the Son of God is Son in name only, and not in nature, is not the faith of the Gospels and of the Apostles”,  This is what the Church was built on and anyone who believes and teaches as the Apostles taught is part of the One True Church, which includes Roman Catholics, Protestants, Evangelicals and Orthodox Christians.