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
By Universal Affirmative implies Particular Affirmative iff First Predicate is not Vacuous:
 * $\left({\exists y: y = x \land \forall y: \left({y = x \implies P \left({x}\right)}\right)}\right) \implies \exists y: \left({y = x \land P \left({x}\right)}\right)$

Then:

Similarly:

The above two statements comprise the other direction of the biconditional assertions.

Together, $(1)$, $(2)$, $(3)$, and $(4)$ prove the two assertions.