Substitution of Elements

Theorem
Let $a$, $b$, and $x$ be sets.


 * $a = b \implies \left({a \in x \iff b \in x}\right)$

Proof
By the Axiom of Extension:
 * $a = b \implies \left({a \in x \implies b \in x}\right)$

Equality is Symmetric, so also by the Axiom of Extension:
 * $a = b \implies \left({b \in x \implies a \in x}\right)$