During the Lord’s earthly ministry Peter experienced a privileged position among the apostles.
He was one of the first apostles called by Jesus (Matt 4:18, 10:2).
He was given a special name by the Lord (remember his real name was Simon, John 1:42).
He was given special revelations from God and was given the keys of the kingdom of heaven (Matt 16:16-19).
He was part of the exclusive group privy to experiencing the transfigured Lord in his kingdom glory on the mount (Matt 17:1-5).
A special prayer was proffered on his behalf to deliver him from Satan and to strengthen his faith (Luke 22:31,32).
Along with James and John he was chosen to keep watch that night in Gethsemane (Matt 26:35-36).
He was instructed thrice personally after the resurrection to feed the lambs of Christ (John 21:15).
On numerous occasions he speaks on behalf of the group of apostles and in the early chapters of Acts he clearly displays his unique leadership authority in choosing the replacement for Judas, speaking to the crowds at Pentecost, and directing the affairs of the Pentecostal church.
However, later in Acts, after Paul receives his divine apostleship and eventually meets with Peter at Jerusalem, it is recognized that the same power that worked in Peter worked mightily also in Paul:
“For he that wrought effectually in Peter to the apostleship of the circumcision, the same was mighty in me toward the Gentiles:” – Galatians 2:8
In this historic event at the Jerusalem council, the apostles grant Paul his due authority as God had given it as the dispenser of a special revelation.
Therefore when Peter, who had been the chiefest of the apostles, walks contrary to Paul’s authoritative message, Paul withstands him to the face.
“But when Peter was come to Antioch, I withstood him to the face, because he was to be blamed. ” – Galatians 2:11
It is at this point that Paul displays his authority over Peter and defends the ‘truth of the gospel’ even against hypocrisy among his own brethren.
Do you suppose it is true what Paul writes to the Corinthians?
“For I suppose I was not a whit behind the very chiefest apostles” – 2 Corinthians 11:5