Theorem absneu

Description: Restricted existential uniqueness determined by a singleton. (Contributed by NM, 29-May-2006)

	Ref	Expression
Assertion	absneu	$$⊢\left(A\in V\wedge \left\{x|\phi \right\}=\left\{A\right\}\right)\to \exists !x\phi$$

Proof

Step	Hyp	Ref	Expression
1		sneq	$$⊢y=A\to \left\{y\right\}=\left\{A\right\}$$
2	1	eqeq2d	$$⊢y=A\to \left(\left\{x|\phi \right\}=\left\{y\right\}\leftrightarrow \left\{x|\phi \right\}=\left\{A\right\}\right)$$
3	2	spcegv	$$⊢A\in V\to \left(\left\{x|\phi \right\}=\left\{A\right\}\to \exists y\left\{x|\phi \right\}=\left\{y\right\}\right)$$
4	3	imp	$$⊢\left(A\in V\wedge \left\{x|\phi \right\}=\left\{A\right\}\right)\to \exists y\left\{x|\phi \right\}=\left\{y\right\}$$
5		euabsn2	$$⊢\exists !x\phi \leftrightarrow \exists y\left\{x|\phi \right\}=\left\{y\right\}$$
6	4 5	sylibr	$$⊢\left(A\in V\wedge \left\{x|\phi \right\}=\left\{A\right\}\right)\to \exists !x\phi$$