TITLE: Inference rules in ZFC
QUESTION [5 upvotes]: I'm relatively new to formal logic.  I have found a list of ZFC axioms on Wikipedia, but do not know what the rules of inference are.  Is there a resource for what these inference rules are, or could someone list them for me?  I have a copy of Mendelson's text on formal logic, so I can consult that for further reading once I know what ZFC's inference rules are.

REPLY [3 votes]: ZFC is a first-order theory with a classical first-order logic with identity. If textbooks sometimes don't specify a particular system of logic for ZFC it is because it doesn't matter so long as it is a full classical system. You could use the ugly system in Mendelson's book, a natural deduction system, a sequent calculus, a tableaux system ... it won't matter because if there is a proof from the ZFC axioms to the conclusion $\phi$ in one system,  there will be a proof in the others. So just assume your favourite presentation of first-order logic is in play, and off you go ...