Theorem r19.36vf 39324

Description: Restricted quantifier version of one direction of 19.36 2098. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypothesis

Ref	Expression
r19.36vf.1	|- F/ x ps

Assertion

Ref	Expression
r19.36vf	|- ( E. x e. A ( ph -> ps ) -> ( A. x e. A ph -> ps ) )

Proof of Theorem r19.36vf

Step	Hyp	Ref	Expression
1		r19.35 3084	. 2 |- ( E. x e. A ( ph -> ps ) <-> ( A. x e. A ph -> E. x e. A ps ) )
2		r19.36vf.1	. . . 4 |- F/ x ps
3		idd 24	. . . 4 |- ( x e. A -> ( ps -> ps ) )
4	2, 3	rexlimi 3024	. . 3 |- ( E. x e. A ps -> ps )
5	4	imim2i 16	. 2 |- ( ( A. x e. A ph -> E. x e. A ps ) -> ( A. x e. A ph -> ps ) )
6	1, 5	sylbi 207	1 |- ( E. x e. A ( ph -> ps ) -> ( A. x e. A ph -> ps ) )

Syntax hints:   -> wi 4   F/wnf 1708   e. wcel 1990   A.wral 2912   E.wrex 2913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710  df-ral 2917  df-rex 2918

This theorem is referenced by:  iinssf  39327