Substitutivity of Equality

Theorem
Let $x$ and $y$ be sets.

Let $P(x)$ be a well-formed formula of the language of set theory.

Let $P(y)$ be the same proposition $P(x)$ with all free instances of $x$ replaced with free instances of $y$.

Let $=$ denote set equality


 * $x = y \implies ( P(x) \iff P(y) )$

Proof
By induction on the well-formed parts of $P(x)$.