Theorem reusn

Description: A way to express restricted existential uniqueness of a wff: its restricted class abstraction is a singleton. (Contributed by NM, 30-May-2006) (Proof shortened by Mario Carneiro, 14-Nov-2016)

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

Proof

Step	Hyp	Ref	Expression
1		euabsn2	$$⊢\exists !x\left(x\in A\wedge \phi \right)\leftrightarrow \exists y\left\{x|\left(x\in A\wedge \phi \right)\right\}=\left\{y\right\}$$
2		df-reu	$$⊢\exists !x\in A\phi \leftrightarrow \exists !x\left(x\in A\wedge \phi \right)$$
3		df-rab	$$⊢\left\{x\in A|\phi \right\}=\left\{x|\left(x\in A\wedge \phi \right)\right\}$$
4	3	eqeq1i	$$⊢\left\{x\in A|\phi \right\}=\left\{y\right\}\leftrightarrow \left\{x|\left(x\in A\wedge \phi \right)\right\}=\left\{y\right\}$$
5	4	exbii	$$⊢\exists y\left\{x\in A|\phi \right\}=\left\{y\right\}\leftrightarrow \exists y\left\{x|\left(x\in A\wedge \phi \right)\right\}=\left\{y\right\}$$
6	1 2 5	3bitr4i	$$⊢\exists !x\in A\phi \leftrightarrow \exists y\left\{x\in A|\phi \right\}=\left\{y\right\}$$