Theorem rexlimdva2 2651

Description: Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by Glauco Siliprandi, 2-Jan-2022.)

Hypothesis

Ref	Expression
rexlimdva2.1	|- ( ( ( ph /\ x e. A ) /\ ps ) -> ch )

Assertion

Ref	Expression
rexlimdva2	|- ( ph -> ( E. x e. A ps -> ch ) )

Distinct variable groups:   ch, x   ph, x

Allowed substitution hints:   ps( x)   A( x)

Proof of Theorem rexlimdva2

Step	Hyp	Ref	Expression
1		rexlimdva2.1	. . 3 |- ( ( ( ph /\ x e. A ) /\ ps ) -> ch )
2	1	exp31 364	. 2 |- ( ph -> ( x e. A -> ( ps -> ch ) ) )
3	2	rexlimdv 2647	1 |- ( ph -> ( E. x e. A ps -> ch ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2200   E.wrex 2509

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 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-ial 1580  ax-i5r 1581

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-ral 2513  df-rex 2514

This theorem is referenced by:  ctssdclemn0  7300  ctssdc  7303  suplocexprlemru  7929  suplocexprlemloc  7931  suplocsrlemb  8016  aptap  8820  4sqlemffi  12959  4sqleminfi  12960  4sqexercise2  12962  4sqlemsdc  12963  ennnfonelemhom  13026  gsumfzval  13464  innei  14877  ivthinclemlr  15351  ivthinclemur  15353  limccnpcntop  15389  limccoap  15392  2lgslem1c  15809  2lgslem3a1  15816  2lgslem3b1  15817  2lgslem3c1  15818  2lgslem3d1  15819  umgrnloop  15957