Theorem r19.35-1 2460

Description: Restricted quantifier version of 19.35-1 1515. (Contributed by Jim Kingdon, 4-Jun-2018.)

Assertion

Ref	Expression
r19.35-1	|- ( E. x e. A ( ph -> ps ) -> ( A. x e. A ph -> E. x e. A ps ) )

Proof of Theorem r19.35-1

Step	Hyp	Ref	Expression
1		r19.29 2450	. . 3 |- ( ( A. x e. A ph /\ E. x e. A ( ph -> ps ) ) -> E. x e. A ( ph /\ ( ph -> ps ) ) )
2		pm3.35 329	. . . 4 |- ( ( ph /\ ( ph -> ps ) ) -> ps )
3	2	reximi 2416	. . 3 |- ( E. x e. A ( ph /\ ( ph -> ps ) ) -> E. x e. A ps )
4	1, 3	syl 14	. 2 |- ( ( A. x e. A ph /\ E. x e. A ( ph -> ps ) ) -> E. x e. A ps )
5	4	expcom 109	1 |- ( E. x e. A ( ph -> ps ) -> ( A. x e. A ph -> E. x e. A ps ) )

Syntax hints:   -> wi 4   /\ wa 97   A.wral 2306   E.wrex 2307

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427

This theorem depends on definitions:  df-bi 110  df-ral 2311  df-rex 2312

This theorem is referenced by:  r19.36av  2461  r19.37  2462  iinexgm  3908  bndndx  8180