Theorem rexsng

Description: Restricted existential quantification over a singleton. (Contributed by NM, 29-Jan-2012) (Proof shortened by AV, 7-Apr-2023)

	Ref	Expression
Hypothesis	ralsng.1	|- ( x = A -> ( ph <-> ps ) )
Assertion	rexsng	|- ( A e. V -> ( E. x e. { A } ph <-> ps ) )

Proof

Step	Hyp	Ref	Expression
1		ralsng.1	|- ( x = A -> ( ph <-> ps ) )
2		nfv	|- F/ x ps
3	2 1	rexsngf	|- ( A e. V -> ( E. x e. { A } ph <-> ps ) )