Theorem euabsn2

Description: Another way to express existential uniqueness of a wff: its class abstraction is a singleton. (Contributed by Mario Carneiro, 14-Nov-2016)

	Ref	Expression
Assertion	euabsn2	$$⊢\exists !x\phi \leftrightarrow \exists y\left\{x|\phi \right\}=\left\{y\right\}$$

Proof

Step	Hyp	Ref	Expression
1		eu6	$$⊢\exists !x\phi \leftrightarrow \exists y\forall x\left(\phi \leftrightarrow x=y\right)$$
2		absn	$$⊢\left\{x|\phi \right\}=\left\{y\right\}\leftrightarrow \forall x\left(\phi \leftrightarrow x=y\right)$$
3	2	exbii	$$⊢\exists y\left\{x|\phi \right\}=\left\{y\right\}\leftrightarrow \exists y\forall x\left(\phi \leftrightarrow x=y\right)$$
4	1 3	bitr4i	$$⊢\exists !x\phi \leftrightarrow \exists y\left\{x|\phi \right\}=\left\{y\right\}$$