First-order set theory is often introduced with $\in$ as its relevant, nonlogical symbol and $=$ interpreted as the regular equality and thus immediately satisfying the $I_\text{1-4}$ axioms (reflexivity, symmetry, transitivity, substitution) in whichever structure it may be found. We could however define the $=$ predicate by set-theoretic extensionality, i.e. $x = y \equiv \forall z \ (z \in x \leftrightarrow z \in y)$. If one were to do this, how would one go about inductively proving the (metatheoretic) statement of $I_4$. The other $I_\text{1-3}$ seem simple enough to prove, but I'm a tad stumped as to how exactly one should start with the base case with regard to atomic formulae.
Ideally one should first prove that for all atomic formulae $x_i \in x_j$ $I_4$ indeed holds, then move on (inductively) to the $\phi_1 \land \phi_2$, $\neg \phi_1$, $\forall x \ \phi_1(x)$, etc. cases. But I don't exactly know what to "prove" when it comes to such atomic formulae! Say I consider two arbitrary $x$ and $y$ and suppose that $x = y$. Which related atomic formulae am I to consider now? Formulae of the form "$x \in z$"/"$y \in z$"? for any arbitrary $z$? Are the formulae I consider supposed to be closed? In which case, should I instead commence with something like $\forall z \ (x \in z)$ and prove that $\forall z \ (y \in z)$ also holds? I'm confused as to the beginning steps of such an inductive proof.
