Substitutivity of Equality

Theorem
Let $\phi(x)$ be a proposition that contains $x$ as a free variable. Let $\phi(y)$ be the same proposition $\phi(x)$ with all free instances of $x$ replaced with free instances of $y$.

$y$ must not be a free variable in $\phi(x)$, and $x$ must not be a free variable in $\phi(y)$. Then:


 * $\displaystyle x = y \implies ( \phi(x) \iff \phi(y) )$

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