The only mention of Peter in Antioch is Galatians 2:11 and in Galatians 2:12 he demonstrates his failings again by cowering in terror from those who came from Jerusalem and James (not the Apostle) and belonged to the circumcision group. Further on in verse 14 Paul had to kick his butt so he would "get with the program". Yeah....this man was the first something....but it was not the Bishop.
Again, your tradition is not backed up by scripture.....or reality.
That is confusing. Another poster along with the Catholic church says that Peter was the first Biship of Rome. Peter himself does not claim either office but only identifies himself as an elder in the church (I Peter 5:1) There is a lot of misunderstanding and personal opinion posted on this thread.