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http://www.vatican.va/holy_father/leo_xiii/encyclicals/documents/hf_l-xiii_enc_29061896_satis-cognitum_en.html

from SATIS COGNITUM
ENCYCLICAL OF POPE LEO XIII ON THE UNITY OF THE CHURCH

Jesus Christ, therefore, appointed Peter to be that head of the Church; and He also determined that the authority instituted in perpetuity for the salvation of all should be inherited by His successors, in whom the same permanent authority of Peter himself should continue. And so He made that remarkable promise to Peter and to no one else: "Thou are Peter, and upon this rock I will build my church" (Matt. xvi., 18). "To Peter the Lord spoke: to one, therefore, that He might establish unity upon one" (S. Pacianus ad Sempronium, Ep. iii., n. 11). "Without any prelude He mentions St. Peter's name and that of his father (Blessed art thou Simon, son of John) and He does not wish Him to be called any more Simon; claiming him for Himself according to His divine authority He aptly names him Peter, from petra the rock, since upon him He was about to found His Church" (S. Cyrillus Alexandrinus, In Evang. Joan., lib. ii., in cap. i., v. 42).

The Universal Jurisdiction of St. Peter

12. From this text it is clear that by the will and command of God the Church rests upon St. Peter, just as a building rests on its foundation. Now the proper nature of a foundation is to be a principle of cohesion for the various parts of the building. It must be the necessary condition of stability and strength. Remove it and the whole building falls. It is consequently the office of St. Peter to support the Church, and to guard it in all its strength and indestructible unity.

So He who is Himself "the chief corner-stone in whom all the building being framed together, groweth up in a holy temple in the Lord" (Eph. ii., 21), placed Peter as it were a stone to support the Church. "When he heard `thou art a rock,' he was ennobled by the announcement. Although he is a rock, not as Christ is a rock, but as Peter is a rock. For Christ is by His very being an immovable rock; Peter only through this rock. Christ imparts His gifts, and is not exhausted....He is a priest, and makes priests. He is a rock, and constitutes a rock" (Hom. de Poenitentia, n. 4 in Appendice opp. S. Basilii).