Equivalence of Definitions of Unique Existential Quantifier/Definition 1 iff Definition 2

Proof
Suppose Definition 1, that for some $x$, both:
 * (1) : $\map P x$

and:
 * (2) : $\forall y : \paren { \map P y \implies x = y }$

From (1):
 * $x = y \implies \map P y$

From this and (2), we conclude:
 * $\exists x : \forall y : \paren { \map P y \iff x = y }$

Suppose Definition 2, that for some $x$ and every $y$:
 * $\map P y \iff x = y$

Taking $y = x$ yields:
 * $x = x \implies \map P x$

implying that $\map P x$.

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