Theorem rexlimdva2 2671

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 2667	1 |- ( ph -> ( E. x e. A ps -> ch ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2209   E.wrex 2529

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 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587  ax-i5r 1588

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-ral 2533  df-rex 2534

This theorem is referenced by:  ctssdclemn0  7440  ctssdc  7443  suplocexprlemru  8076  suplocexprlemloc  8078  suplocsrlemb  8163  aptap  8968  4sqlemffi  13153  4sqleminfi  13154  4sqexercise2  13156  4sqlemsdc  13157  ennnfonelemhom  13284  gzsumfzval  13688  innei  15187  ivthinclemlr  15661  ivthinclemur  15663  limccnpcntop  15699  limccoap  15702  2lgslem1c  16123  2lgslem3a1  16130  2lgslem3b1  16131  2lgslem3c1  16132  2lgslem3d1  16133  umgrnloop  16271