Alphabetic Substitution is Semantically Equivalent

Theorem
Let $\map \phi x$ be a WFF of predicate logic.

Let $z$ be free for $x$ in $\phi$.

Let $z$ not occur freely in $\phi$.

Let $\map \phi z$ be the result of the alphabetic substitution of $z$ for $x$.

Then:


 * $\forall x: \map \phi x$ and $\forall z: \map \phi z$ are semantically equivalent
 * $\exists x: \map \phi x$ and $\exists z: \map \phi z$ are semantically equivalent