Theorem ralsn

Description: Convert a universal quantification restricted to a singleton to a substitution. (Contributed by NM, 27-Apr-2009)

	Ref	Expression
Hypotheses	ralsn.1	|- A e. _V
	ralsn.2	|- ( x = A -> ( ph <-> ps ) )
Assertion	ralsn	|- ( A. 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	ralsng	|- ( A e. _V -> ( A. x e. { A } ph <-> ps ) )
4	1 3	ax-mp	|- ( A. x e. { A } ph <-> ps )