The Lord then says, “And I tell you, you are Peter, and on this rock will I build my church.” In St. John 1 [:42], he calls him Cephas, “You shall be called Cephas,” Keph in Hebrew, Kepha in Chaldean, and Petros or Petra in Greek…all of which mean rock… Now the Lord wants to say, “You are Peter, that is, a man of rock. For you have recognized and named the right Man, who is the true rock, as Scripture names him, Christ. On this rock, that is, on me, Christ, I will build all of my Christendom, just as you and the other disciples are built on it through my Father in heaven, who revealed it to you.”

In plain…one would say, “You say (on behalf of all) that I am the Messiah or Christ, the Son of the living God; very well then, I say to you, you are a Christian, and I shall build my church on a Christian.” For…the word “Christ” means both the Lord himself, as one sings, “Christ the Lord is risen, Christ ascended to heaven,” and he who believes in the Lord Christ, as one says, “You are a Christ.” Thus Luke in Acts… says that the disciples in Antioch were first called Christians, which is why names have survived such as, “Christians, Christendom, Christian faith,” etc. So here our Lord gives Simon, son of Jona, the name “man of rock” or “Christian” because he, from the Father, recognized the rock, or Christ, and praised him with his mouth on behalf of all the apostles.

From this it is clear enough that by the building of his church on the rock or on himself, Christ meant nothing else but (as was said above, from the apostles Peter and Paul) the common Christian faith, that whoever believes in Christ is built on this rock and will attain salvation, even against all the gates of hell…This is the simple, single, certain understanding of these words….[ii]

Namely, that sedevacantists believe the Church is founded on St. Peter's (or his successor's) faith, not on the man himself.

How do sedevacantists answer this objection/accusation?

Quote from: Jehanne

Quote from: Geremia

Namely, that sedevacantists believe the Church is founded on St. Peter's (or his successor's) faith, not on the man himself.

How do sedevacantists answer this objection/accusation?

Sedevacantists hold that the Petrine office is what is permanent and not necessarily the man who claims to occupy it.  Is a President about the Constitution (for US-based members and others)?  Is a Pope above his Magisterium or the author of it?  Just as a President can be a traitor to his office (namely, protecting the Constitution), so, too, a Pope can betray his own Magisterium.  Sedevacantists "salute the position" and not necessarily "the person."

I presume you meant "Is a President above the Constitution" -- interestingly enough, there are different schools of thought on the matter as much as there are different schools of thought on the pope question.