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  7308  ctssdc  7311  suplocexprlemru  7938  suplocexprlemloc  7940  suplocsrlemb  8025  aptap  8829  4sqlemffi  12968  4sqleminfi  12969  4sqexercise2  12971  4sqlemsdc  12972  ennnfonelemhom  13035  gsumfzval  13473  innei  14886  ivthinclemlr  15360  ivthinclemur  15362  limccnpcntop  15398  limccoap  15401  2lgslem1c  15818  2lgslem3a1  15825  2lgslem3b1  15826  2lgslem3c1  15827  2lgslem3d1  15828  umgrnloop  15966