The proof is in the bible: In John 1:42, ‘Jesus looked at him and said, “You are Simon son of John. You will be called Cephas” (which, when translated, is Peter).’
The truth is that there is no dichotomy between Petros and Petra. Petros is not a word in biblical Greek. There exists some Greek poetry and literature which uses a metaphor wherein a mother stone (Petra) gives birth to a quarried stone (Petros), but this distinction was hardly familiar usage; Peter is called Petros, not Petra IN THE GREEK simply because to call him Petra would be to give him a girl’s name. The fact that such a distinction is impossible is made plain by the fact that Jesus actually called him Cephas.
Further, Jesus does NOT say, “You are Petros, and on Petra I will build my church.” Jesus specifically says, “on THIS rock.” On what rock? The use of “THIS” means he is referring to what he has previously mentioned: Peter.
Also, if Jesus MEANT a distinction, he would have used the word, “lithos.” Lithos refers to “*a* stone, as opposed to stone, and is used commonly in the bible. Petros, on the other hand, in all of early Christian literature, is used only to refer to Peter, and never anywhere else, including the entirety of the Greek bible.
Lastly, what does Jesus immediately do? He gives Peter the authority to rule over his church!
The counter-argument is that Jesus refers to faith when he refers to “petra,” but the same Church Fathers who taught this also taught that Peter was the rock! Yes, Jesus chose Peter to be the rock because Peter’s faith was rock. Peter was a sinful man, as popes have been. He erred gravely in his administration of the Church, as Paul boasts of telling him. But it was the doctrinal statement of Peter which was rock, and to this day, that has always been the particular spiritual charism that the Church asserts of the papacy: not that it is inerrant (consider the scandal of Pope Honorius), not that it is impeccable (need we even mention some of the popes’ wickedness?), but that it is infallible when it freely asserts moral doctrine from the Chair of St Peter for the benefit of all Christians.