Theorem rexlimdva2 2663

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 2659	1 |- ( ph -> ( E. x e. A ps -> ch ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2203   E.wrex 2521

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 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-17 1575  ax-ial 1583  ax-i5r 1584

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-ral 2525  df-rex 2526

This theorem is referenced by:  ctssdclemn0  7401  ctssdc  7404  suplocexprlemru  8034  suplocexprlemloc  8036  suplocsrlemb  8121  aptap  8924  4sqlemffi  13094  4sqleminfi  13095  4sqexercise2  13097  4sqlemsdc  13098  ennnfonelemhom  13166  gsumfzval  13604  innei  15028  ivthinclemlr  15502  ivthinclemur  15504  limccnpcntop  15540  limccoap  15543  2lgslem1c  15963  2lgslem3a1  15970  2lgslem3b1  15971  2lgslem3c1  15972  2lgslem3d1  15973  umgrnloop  16111