Actually - St Irenaeus was genuis in adding to the fact not only did Rome contain the succession of the Prince of the Apostles, Peter, but also adding in the Apostle who taught the Gentiles - Paul.
So in a way - the two converging into the Chair only makes the Chair of Peter more significant.
Because the umpteen hundred quotes i have on the Pope

BUT also the fact the most versed Apostle Paul.
However; St Irenaeus lived around the time of 175 AD when he wrote. He was born about 130 AD and was a disciple under St Polycarp who was a Bishop thru St John the Apostle and his friend.
So ya want me to do the leg work eh?
Ok fine..
The two authorities presuppose the following list: (1) Peter, xxv; (2)
Linus, xii; (3) Anencletus [
Cletus], xii; (4) Clement, ix; (5) Evarestus, viii; (6) Alexander, x; (7)
Sixtus, x; (8) Telesophorus, xi; (9)
Hyginus, iv; (10)
Pius, xv; (11)
Anicetus, xi;, (12) Soter, viii; (13) Eleutherius, xv; (14) Victor, x; (15)
Zephyrinus, xviii; (16)
Callistus, v; (17)
Urban, viii; (18)
Pontian, v (Harnack, "Chronologie", I, 152).
St Irenaeus writes [pay attention BECAUSE he states] in the 1st paragraph here that St Peter's line is VERY significant compared to the other's.
2. Since, however, it would be very tedious, in such a volume as this, to reckon up the successions of all the
Churches, we do put to confusion all those who, in whatever manner, whether by an
evil self-pleasing, by vainglory, or by blindness and perverse opinion, assemble in unauthorized meetings; [we do this, I say,] by indicating that tradition derived from the
apostles, of the very great, the very ancient, and universally
known Church founded and organized at
Rome by the two most
glorious apostles, Peter and
Paul; as also [by pointing out] the
faith preached to
men, which comes down to our time by means of the successions of the
bishops.
For it is a matter of necessity that every Church should agree with this Church, on account of its preeminent authority, that is, the faithful everywhere, inasmuch as the
apostolical tradition has been preserved continuously by those [faithful men] who exist everywhere.
3. The blessed
apostles, then, having founded and built up the
Church, committed
into the hands of Linus the office of the
episcopate. Of this Linus,
Paul makes mention in the Epistles to Timothy. To
him succeeded Anacletus; and
after him, in the third place from the
apostles,
Clement was allotted the
bishopric. This man, as he
had seen the blessed apostles, and had been conversant with them, might be said to have the preaching of the apostles still echoing [in his ears], and their traditions before his eyes. Nor was he alone [in this], for there were many still remaining who had received instructions from the
apostles. In the time of this Clement, no small dissension having occurred among the brethren at
Corinth,
the Church in Rome despatched a most powerful letter to the Corinthians, exhorting them to peace, renewing their
faith, and declaring the tradition which it had lately received from the
apostles, proclaiming the one
God,
omnipotent, the Maker of heaven and earth, the Creator of
man, who brought on the deluge, and called
Abraham, who led the people from the land of
Egypt, spoke with
Moses, set forth the law, sent the
prophets, and who has prepared fire for the
devil and his
angels. From this document, whosoever chooses to do so, may learn that He, the Father of
our Lord Jesus Christ, was preached by the
Churches, and
may also understand the apostolical tradition of the Church, since this Epistle is of older date than these men who are now propagating
falsehood, and who conjure into
existence another god beyond the Creator and the Maker of all existing things. To this Clement there
succeeded Evaristus. Alexander followed Evaristus; then,
sixth from the apostles,
Sixtus was appointed; after him,
Telephorus, who was
gloriously martyred;
then Hyginus;
after him, Pius; then
after him, Anicetus. Soter having succeeded Anicetus,
Eleutherius does now,
in the twelfth place from the apostles, hold the inheritance of the episcopate. In this order, and by this succession, the
ecclesiastical tradition from the
apostles, and the preaching of the
truth, have come down to us.
And this is most abundant proof that there is one and the same vivifying faith,
which has been preserved in the Church from the apostles until now, and handed down in truth.
4. But
Polycarp also was not only instructed by
apostles, and conversed with many who had seen Christ, but was also, by
apostles in Asia, appointed
bishop of the Church in
Smyrna, whom I also saw in my early youth, for he tarried [on earth] a very long time, and, when a very old man,
gloriously and most nobly suffering
martyrdom, departed this life,
having always taught the things which he had learned from the apostles, and which the Church has handed down, and which alone are true.
CHURCH FATHERS: Against Heresies, III.3 (St. Irenaeus)