Equality implies Substitution

Theorem
Let $\map P x$ denote a Well-Formed Formula which contains $x$ as a free variable.

Then the following are tautologies:


 * $\forall x: \paren {\map P x \iff \exists y: \paren {y = x \land \map P y} }$
 * $\forall x: \paren {\map P x \iff \forall y: \paren {y = x \implies \map P y} }$

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:
 * $\paren {\exists y: y = x \land \forall y: \paren {y = x \implies \map P x} } \implies \exists y: \paren {y = x \land \map P x}$

Then:

Similarly:

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

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