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Apostolic succession is evident in the first missionary journey of Paul and Barnabas. They "appointed elders [bishops and priests] for them in every church, with prayers and fasting, they committed them to the Lord in whom they believe" (Acts 14:23).
In his second letter to Timothy St. Paul laid out the generational program for apostolic succession that was practiced by the Apostles and their successors, and is continued to the present time: "You then, my son, be strong in the grace that is in Christ Jesus, and what you [Timothy was a second generation of Christians] have heard from me [Paul was of the first generation] before many witnesses entrust to the faithful men [the third generation] who will be able to teach others [the fourth generation] also (2 Tim 2:1-2).
By the end of the second century, apostolic succession was understood as the sure indicator of orthodoxy. St. Irenaeus of Lions, writing against the Gnostics around the year 180, affirmed "the tradition of the Apostles," was safeguarded in the unbroken line of succession of those men who were instituted bishops by the Apostles, and their successors. He placed the greatest importance on the successors of St. Peter in Rome.
We can also look at what some fathers of the Church had to say:
“In the city of Rome the episcopal chair was given first to Peter, the chair in which Peter sat, the same who was head--that is why he is also called Cephas [‘Rock’]--of all the apostles, the one chair in which unity is maintained by all. Neither do the apostles proceed individually on their own, and anyone who would [presume to] set up another chair in opposition to that single chair would, by that very fact, be a schismatic and a sinner. . . . Recall, then, the origins of your chair, those of you who wish to claim for yourselves the title of holy Church.”
-St. Optatus, “The Schism of the Donatists,” c. 367 A.D.
“They (the Novatian heretics) have not the succession of Peter, who hold not the chair of Peter, which they rend by wicked schism; and this, too, they do, wickedly denying that sins can be forgiven (by the sacrament of confession) even in the Church, whereas it was said to Peter: 'I will give unto thee the keys of the kingdom of Heaven, and whatsoever thou shalt bind on Earth, shall be bound in Heaven, and whatsoever thou shall loose on Earth shall be loosed in Heaven.'"
-St. Ambrose of Milan, “On Penance,” 388 A.D.