Substitution of Elements

Theorem

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

$a = b \implies \paren {a \in x \iff b \in x}$

Proof

By the Axiom of Extension:

$a = b \implies \paren {a \in x \implies b \in x}$

Equality is Symmetric, so also by the Axiom of Extension:

$a = b \implies \paren {b \in x \implies a \in x}$

$\blacksquare$

Sources

• 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 3.3$