Theorem rexsn

Description: Convert an existential quantification restricted to a singleton to a substitution. (Contributed by Jeff Madsen, 5-Jan-2011)

	Ref	Expression
Hypotheses	ralsn.1	|- A e. _V
	ralsn.2	|- ( x = A -> ( ph <-> ps ) )
Assertion	rexsn	|- ( E. x e. { A } ph <-> ps )

Proof

Step	Hyp	Ref	Expression
1		ralsn.1	|- A e. _V
2		ralsn.2	|- ( x = A -> ( ph <-> ps ) )
3	2	rexsng	|- ( A e. _V -> ( E. x e. { A } ph <-> ps ) )
4	1 3	ax-mp	|- ( E. x e. { A } ph <-> ps )