I know that to prove $A$ is equivalent to $B$, I have to assume $A$ then prove $B$ and then assume $B$ and prove $A$. But let’s say that $C$ is a axiom. Can I use $C$ in my proof of $B$ from $A$ and vice versa? Or can I only use $A$ and $B$, and not $C$?
Edit: I can only prove $(A \land C) \equiv (B\land C)$. Is there any way to prove $A \equiv B$? Remember $C$ is an axiom.