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  7288  ctssdc  7291  suplocexprlemru  7917  suplocexprlemloc  7919  suplocsrlemb  8004  aptap  8808  4sqlemffi  12934  4sqleminfi  12935  4sqexercise2  12937  4sqlemsdc  12938  ennnfonelemhom  13001  gsumfzval  13439  innei  14852  ivthinclemlr  15326  ivthinclemur  15328  limccnpcntop  15364  limccoap  15367  2lgslem1c  15784  2lgslem3a1  15791  2lgslem3b1  15792  2lgslem3c1  15793  2lgslem3d1  15794  umgrnloop  15931