Ah, so my question was more along the line: does finite axiomatizability of a stronger (consistent) theory imply finite axiomatizability of the weaker theory?
If I’m not mistaken, NBG and ZFC are a counterexample to this: NBG is a conservative extension of ZFC (and therefore stronger than ZFC), but NBG is finitely axiomatizable while ZFC is not.
If I’m not mistaken, NBG and ZFC are a counterexample to this: NBG is a conservative extension of ZFC (and therefore stronger than ZFC), but NBG is finitely axiomatizable while ZFC is not.