If you want to and need to believe those writings, so be it. There were five equal Bishooprics following the death of the last Apostle. There was no Petrine authority. That was fabricated to give the Rome Bishopric top authority. That was only achieved when the Emperor of Rome appointed Leo as ‘The Primate of all Bishops.’ That’s real History as unraveled from documentations not authored by persons with an agenda to slant things toward Rome. BY the way, are you familiar with one of the greatest forgeries in History? The Isodorian Decretals? That was one of your Roman empowerment schemes revealed. But some will follow the traditions of men regardless of what is True.
I believe you are referring to Pentarchy or the Five Patriarchs: Rome, Constantinople, Alexandria, Antioch and Jerusalem. But this is an idea that did not develop until 6th century. Originally a city of any size had its own bishop. Constantinople was not even recognized as a patriarchal see until the 4th century.
There was no Petrine authority. That was fabricated to give the Rome Bishopric top authority. That was only achieved when the Emperor of Rome appointed Leo as ‘The Primate of all Bishops.’
Pope St. Leo I was in the 5th century. Note that Augustine was speaking of Petrine authority prior to this. Around A.D. 80 Pope St. Clement gives instruction to the church in Corinth. In the 1st century Pope Victor I exercised his authority over the church in mandating that the eastern bishops accept the date for Easter, threatening them with excommunication.
Have you read “Eusibius: Church History”?