Leibniz's Law for Sets

Theorem
Let $S$ be an arbitrary set.

Then:


 * $x = y \dashv \vdash x \in S \iff y \in S$

for all $S$ in the universe of discourse.

This is therefore the justification behind the notion of the definition of set equality.

Proof
A direct application of Leibniz's law.

Also see

 * Axiom:Axiom of Extensionality