St. Peter's successors carried on his office, the importance of which grew with the growth of the 
Church. In 97 serious dissensions troubled the 
Church of 
Corinth. The Roman 
Bishop, Clement, unbidden, wrote an authoritative letter to restore peace. St. John was still living at Ephesus, yet neither he nor his interfered with 
Corinth. Before 117 
St. Ignatius of Antioch addresses the 
Roman Church as the one which "presides over charity . . . which has never deceived any one, which has taught others." 
St. Irenæus (180-200) states the theory and practice of 
doctrinal unity as follows: 
With this 
Church [of 
Rome] because of its more powerful principality, every 
Church must agree, that is the faithful everywhere, in this [i.e. in communion with the 
Roman Church] the tradition of the 
Apostles has ever been preserved by those on every side. (Adv. Haereses, III)
The 
heretic Marcion, the 
Montanists from Phrygia, 
Praxeas from 
Asia, come to 
Rome to gain the countenance of its 
bishops; 
St. Victor, 
Bishop of Rome, threatens to 
excommunicate the 
Asian Churches; 
St. Stephen refuses to receive 
St. Cyprian's deputation, and separates himself from various Churches of the East; Fortunatus and Felix, 
deposed by 
Cyprian, have recourse to 
Rome; 
Basilides, 
deposed in 
Spain, betakes himself to 
Rome; the 
presbyters of Dionysius, 
Bishop of 
Alexandria, complain of his 
doctrine to Dionysius, 
Bishop of Rome; the latter expostulates with him, and he explains. The fact is indisputable: the 
Bishops of 
Rome took over Peter's Chair and Peter's office of continuing the work of Christ [Duchesne, "The 
Roman Church before Constantine", 
Catholic Univ. Bulletin (October, 1904) X, 429-450]. To be in continuity with the 
Church founded by Christ affiliation to the See of Peter is 
necessary, for, as a matter of history, there is no other 
Church linked to any other 
Apostle by an unbroken chain of successors. Antioch, once the 
see and centre of St. Peter's labours, fell into the hands of 
Monophysite patriarchs under the Emperors Zeno and Anastasius at the end of the fifth century. The 
Church of Alexandria in 
Egypt was founded by 
St. Mark the 
Evangelist, the mandatory of St. Peter. It flourished exceedingly until the 
Arian and 
Monophysite heresies took root among its people and gradually led to its extinction. The shortest-lived 
Apostolic Church is that of 
Jerusalem. In 130 the Holy City was destroyed by Hadrian, and a new town, Ælia Capitolina, erected on its site. The new 
Church of Ælia Capitolina was subjected to Caesarea; the very name of 
Jerusalem fell out of use till after the 
Council of Nice (325). 
http://www.newadvent.org/cathen/01641a.htm
So much for the Jerusalem-as-centre theory.