How to use axiom of extensionality

Question

Assume $x = y$ and we want to proof $\{x\} = \{y\}$. I read that this follows from the axiom of extensionality. The axiom states: $\forall x \forall y (\forall z( z \in x \iff z \in y) \implies x = y)$. Obviously $z \in \{x\} \iff z = x \iff z = y \iff z \in \{y\}$ so $\{x\} = \{y\}$. Am I doing this correctly or am I assuming things I can't directly assume? For example do I need to proof that $z = x \iff z = y$ with axioms or is this just a consequence of substitution.