Is it possible to prove that equal power sets imply equal sets using the ZF axioms?
Well, yes, Zermelo–Fraenkel set theory has only two predicates: equality and membership. However, $\displaystyle A\subseteq B$ is easily expressible as $\displaystyle \forall x.\,x\in A\to x\in B$. Subset relation is used in the axiom of power set.