Equivalence of Definitions of Unique Existential Quantifier/Definition 2 iff Definition 3

Proof
Suppose Definition 2, that for some $x$:
 * (1) : $\forall y : \paren { \map P y \iff x = y }$

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

implying that $\exists x : \map P x$.

Suppose $\map P y$ and $\map P z$ for arbitrary $y$ and $z$.

Then from (1), $y = x$ and $z = x$, giving:
 * $\forall y : \forall z : \paren { \paren { \map P y \land \map P z } \implies y = z }$

Suppose Definition 3, that:
 * (1) : $\exists x : \map P x$

and for arbitrary $y$ and $z$:
 * (2) : $\paren { \map P y \land \map P z } \implies y = z$

From (2), take $z = x$:
 * $\paren { \map P y \land \map P x } \implies y = x$

Thus, by (1):
 * $\map P y \implies x = y$

Suppose $x = y$.

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

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