Is there any evidence to support the claim that the Apostle St. Peter founded the Church in Antioch and, if so, what are the implications?

According to tradition, the Apostle Peter founded the Church in Antioch in A.D. 34 and is regarded as the first Bishop of Antioch. Here are a few extracts from sources I found regarding this claim: >Peter preached the Gospel in Judea, founded the Church of Antioch and finally came to Rome... St. Peter traditionally is regarded as the leader of the Twelve Disciples of Jesus... In the course of his missionary journeys, Peter founded the Church in Antioch, where the followers of Jesus were first called Christians. St. Peter is regarded by the Church as the first Bishop of Antioch, and the present-day Patriarch of Antioch is his successor in that Apostolic See. - [Saints Peter and Paul](http://ww1.antiochian.org/saints-peter-and-paul) >Church tradition maintains that the See of Antioch was founded by Saint Peter the Apostle in A.D. 34... It was the resolution of this conflict [male circumcision] at the Council of Jerusalem under Saint James the Apostle that determined the direction of the Antiochian mission to the Gentiles, and the dynamic nature of that Christian community in its missionary outreach. - [The Patriarchate of Antioch: Founded by Saints Peter and Paul](http://ww1.antiochian.org/patofant) >Patriarch of Antioch is a traditional title held by the Bishop of Antioch (founded by Saint Peter). As the traditional "overseer" (ἐπίσκοπος, episkopos, from which the word bishop is derived) of the first gentile Christian community, the position has been of prime importance in the church from its earliest period. This diocese is one of the few for which the names of its bishops from the apostolic beginnings have been preserved... According to church tradition, this ancient Patriarchate was founded by the Apostle Saint Peter. - [Patriarch of Antioch (Wikipedia)](https://en.wikipedia.org/wiki/Patriarch_of_Antioch) Apparently Ignatius was bishop of Antioch (martyred under Emperor Trajan circa A.D. 110) Theophilus was bishop of Antioch (A.D. 169-183) Serapion was bishop of Antioch (A.D. 190-203) Is there any substantive evidence to support the claim by the Antiochian Orthodox Church that the apostle Peter founded their church? And, if so, what are the implications with regard to the Roman Catholic Church claiming Peter founded their church in Rome?