Was Fermat's last theorem proved based on Peano's postulates? Is the proof of Fermat's last theorem solely based on the Peano's postulates $+$ first order logic? Or it contains other axiomatic systems as well? What does it mean from foundations of math perspective to use several axiomatic systems to prove a conjecture? Do we know these axiomatic systems are consistent with one another? I'm not sure if I am asking it the right way, but I think logicians only prove the consistency of the axioms of one system not two different systems.
Logicians also prove the consistency of a formal system relative to each other. For instance, if ZFC is consistent, then PA (Peano arithmetic) is consistent. So in that sense ZFC is stronger than PA.
Fermat's last theorem was proved in a system stronger that PA. In that sense, it might be the case that the proof is inconsistent (we'll likely never know) while PA is still consistent. Plus is still unclear if Fermat's last theorem could be proved only using PA (it might be undecidable on PA but I am not aware of that).
Logicians also prove the consistency of a formal system relative to each other. For instance, if ZFC is consistent, then PA (Peano arithmetic) is consistent. So in that sense ZFC is stronger than PA.
Fermat's last theorem was proved in a system stronger that PA. In that sense, it might be the case that the proof is inconsistent (we'll likely never know) while PA is still consistent. Plus is still unclear if Fermat's last theorem could be proved only using PA (it might be undecidable on PA but I am not aware of that).