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

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

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:
 * $(1): \quad \exists x : \map P x$

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

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