Equality implies Substitution

Theorem
Let $P\left({ x }\right)$ denote a Well-Formed Formula which contains $x$ as a free variable.

Then the following are tautologies:


 * $\forall x: \left({ P\left({ x }\right) \iff \exists y: \left({ y = x \land P\left({ y }\right) }\right) }\right)$
 * $\forall x: \left({ P\left({ x }\right) \iff \forall y: \left({ y = x \implies P\left({ y }\right) }\right) }\right)$

Note that when $y$ is substituted for $x$ in either formula, it is false in general; compare Confusion of Bound Variables.

Proof

 * $\left({ \exists y: y = x \land \forall y: \left({ y = x \implies P\left({ x }\right) }\right) }\right) \implies \exists y: \left({ y = x \land P\left({ x }\right) }\right)$ by Introduction of Conjunct into Existential Quantifier

Similarly:

The above two statements comprise the other direction of the biconditional assertions. Together, (1), (2), (3), and (4) prove the two assertions.