Theorem r19.36vf

Description: Restricted quantifier version of one direction of 19.36 . (Contributed by Glauco Siliprandi, 23-Oct-2021)

	Ref	Expression
Hypothesis	r19.36vf.1	|- F/ x ps
Assertion	r19.36vf	|- ( E. x e. A ( ph -> ps ) -> ( A. x e. A ph -> ps ) )

Proof

Step	Hyp	Ref	Expression
1		r19.36vf.1	|- F/ x ps
2		r19.35	|- ( E. x e. A ( ph -> ps ) <-> ( A. x e. A ph -> E. x e. A ps ) )
3		idd	|- ( x e. A -> ( ps -> ps ) )
4	1 3	rexlimi	|- ( E. x e. A ps -> ps )
5	4	imim2i	|- ( ( A. x e. A ph -> E. x e. A ps ) -> ( A. x e. A ph -> ps ) )
6	2 5	sylbi	|- ( E. x e. A ( ph -> ps ) -> ( A. x e. A ph -> ps ) )