According to Book VII, Chapter 30 of Eusebius’ Ecclesiastical History the local churches surrounding Antioch wrote in a letter that they condemned Paul of Samosata (Antioch’s Bishop) and appointed a new one (“we have appointed Domnus.”) They sought both Alexandria’s and Rome’s recognition, similar to how Pope Cornelius sought the world’s approval when Novatian was making the claim at the same time that he was in fact Pope.
In the same letter (or right afterwards, it is hard to tell from how it is written by Eusebius) a most curious detail is recorded, that being, the first ever Roman-government intervention in an affair of the Church: “But as Paul refused to surrender the church building, the Emperor Aurelian was petitioned; and he decided the matter most equitably, ordering the building to be given to those to whom the bishops of Italy and of the city of Rome should adjudge it.”
As we can see, the Pope did not decide the issue of whether Paul stayed in his church building. Rather, the Pope and other Italian Bishops made the decision. This appears not so much as proof of Aurelian’s Papalist or Orthodox ecclesiology, but rather him seeking the input of Christian suffragans in his own backyard.
As for Antioch, it is clear from their letter that they thought that they had already decided the issue. Nevertheless, the Roman Emperor in effect created a legal precedent that Rome and Italian Bishoprics acted as a court of ecclesiastical appeals. We saw this happen again during the Donatist controversy at the behest of Constantine. This became codified in the Council of Serdicia.
Paul of Samosata unintentionally triggered another massively important Roman precedent. In his debate with Paul, Saint Dionysus of Alexandria made Christologically suspect claims and local Egyptian Bishops sought recourse. For the simple reason that Antioch’s Bishop was a heretic, an appeal was made to Rome to settle their theological dispute. Rome in effect acted as a court of theological appeals below that of a church-wide council (which probably was not conceived at that time.)
Hence, by the late third century we saw Rome finally get its first taste of de facto primacy, as only decades before during the Easter and rebaptism controversies Rome’s primacy was received by the Church as no more than symbolic and by way of example–with no actual judicial power or anything of the like. Interestingly enough, it appears to be state power, as opposed to Apostolic Christian practice, to have been the main catalyst in this change. Canon 3 of Constantinople I and Canon 28 of Chalcedon appear to take this for granted, though it should be noted that the Christian world did not universally recognize these canons, though not necessarily for Papist reasons.