Theorem rexlimdva2 2653

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 2649	1 |- ( ph -> ( E. x e. A ps -> ch ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2202   E.wrex 2511

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 1495  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-4 1558  ax-17 1574  ax-ial 1582  ax-i5r 1583

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-ral 2515  df-rex 2516

This theorem is referenced by:  ctssdclemn0  7309  ctssdc  7312  suplocexprlemru  7939  suplocexprlemloc  7941  suplocsrlemb  8026  aptap  8830  4sqlemffi  12987  4sqleminfi  12988  4sqexercise2  12990  4sqlemsdc  12991  ennnfonelemhom  13054  gsumfzval  13492  innei  14906  ivthinclemlr  15380  ivthinclemur  15382  limccnpcntop  15418  limccoap  15421  2lgslem1c  15838  2lgslem3a1  15845  2lgslem3b1  15846  2lgslem3c1  15847  2lgslem3d1  15848  umgrnloop  15986