Definition:Existential Quantifier/Unique/Definition 3

Definition
There exists a unique object $x$ such that $\map P x$, denoted $\exists ! x: \map P x$, both:
 * $\exists x : \map P x$

and:
 * $\forall y : \forall z : \paren {\paren {\map P y \land \map P z} \implies y = z }$

Also see

 * Equivalence of Definitions of Unique Existential Quantifier