OK:
Let me break it down for you.
1. Antioch, by ROMAN belief should have primacy, as St. Peter was Bishop THERE first. (NOT ORTHODOX)
2. Primus inter pares was given to the Bishop of the IMPERIAL CITY. Rome was the Imperial Capitol of the empire, so that title had nothing to do with Rome’s claim of primacy by virtue of Peter. Later, the title was given to the Ecumenical Patriarch of Constantinople.
Last, but not least, my point in all this is that the doctrine of Peterine Primacy is an invention of the Western, or Latin Church, mainly based on the spurious writings of Clement, and that other than being the “First among EQUALS”, there was no universal authority conferred on Rome.