Equivalence of Definitions of Unique Existential Quantifier/Definition 1 iff Definition 3

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

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

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

From $(2)$, there is an $x$ such that $x = y$ and $x = z$.

Thus, for arbitrary $y$ and $z$:
 * $\paren {\map P y \land \map P z} \implies y = z$

and from $(1)$:
 * $\exists x : \map P x$

Suppose Definition 3, that there is an $x$ such that:
 * $(1): \quad \map P x$

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

Taking $z = x$, we have from $(2)$:
 * $\paren {\map P y \land \map P x} \implies y = x$

Thus, from $\map P x$ in $(1)$:
 * $\forall y : \paren {\map P y \implies x = y}$

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