Theorem r19.35-1 2627

Description: Restricted quantifier version of 19.35-1 1624. (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 2614	. . 3 |- ( ( A. x e. A ph /\ E. x e. A ( ph -> ps ) ) -> E. x e. A ( ph /\ ( ph -> ps ) ) )
2		pm3.35 347	. . . 4 |- ( ( ph /\ ( ph -> ps ) ) -> ps )
3	2	reximi 2574	. . 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 116	1 |- ( E. x e. A ( ph -> ps ) -> ( A. x e. A ph -> E. x e. A ps ) )

Syntax hints:   -> wi 4   /\ wa 104   A.wral 2455   E.wrex 2456

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-ial 1534

This theorem depends on definitions:  df-bi 117  df-ral 2460  df-rex 2461

This theorem is referenced by:  r19.36av  2628  r19.37  2629  iinexgm  4156  bndndx  9178