TITLE: Finite axiomatization in logic with equality versus without equality
QUESTION [5 upvotes]: Is there a theory $T$ that can be finitely axiomatized in first-order logic with equality, which can also be axiomatized in first-order logic without equality, but not finitely axiomatized in first-order logic without equality?

REPLY [3 votes]: No: by the compactness theorem, if $T$ and $T'$ are equivalent theories and $T$ is finite, then a finite subset of $T'$ is equivalent to $T$ and $T'$. (Proof: Let $\varphi$ be the conjunction of the finitely many sentences in $T$. Then $T'\cup \{\lnot\varphi\}$ is inconsistent. By compactness, a finite subset of $T'$ implies $\varphi$ and hence implies all of $T'$.)
So taking $T$ to be a finite theory in first-order logic with equality and $T'$ to be an equivalent theory which does not use $=$, it follows that $T'$ is finitely axiomatizable in first-order logic without equality (by a finite subset of $T'$).