Theorem rexsns

Description: Restricted existential quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015) (Revised by NM, 22-Aug-2018)

	Ref	Expression
Assertion	rexsns	|- ( E. x e. { A } ph <-> [. A / x ]. ph )

Proof

Step	Hyp	Ref	Expression
1		velsn	|- ( x e. { A } <-> x = A )
2	1	anbi1i	|- ( ( x e. { A } /\ ph ) <-> ( x = A /\ ph ) )
3	2	exbii	|- ( E. x ( x e. { A } /\ ph ) <-> E. x ( x = A /\ ph ) )
4		df-rex	|- ( E. x e. { A } ph <-> E. x ( x e. { A } /\ ph ) )
5		sbc5	|- ( [. A / x ]. ph <-> E. x ( x = A /\ ph ) )
6	3 4 5	3bitr4i	|- ( E. x e. { A } ph <-> [. A / x ]. ph )