Theorem rexlimdvaa 2595

Description: Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by Mario Carneiro, 15-Jun-2016.)

Hypothesis

Ref	Expression
rexlimdvaa.1	|- ( ( ph /\ ( x e. A /\ ps ) ) -> ch )

Assertion

Ref	Expression
rexlimdvaa	|- ( ph -> ( E. x e. A ps -> ch ) )

Distinct variable groups:   ph, x   ch, x

Allowed substitution hints:   ps( x)   A( x)

Proof of Theorem rexlimdvaa

Step	Hyp	Ref	Expression
1		rexlimdvaa.1	. . 3 |- ( ( ph /\ ( x e. A /\ ps ) ) -> ch )
2	1	expr 375	. 2 |- ( ( ph /\ x e. A ) -> ( ps -> ch ) )
3	2	rexlimdva 2594	1 |- ( ph -> ( E. x e. A ps -> ch ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2148   E.wrex 2456

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 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534  ax-i5r 1535

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-ral 2460  df-rex 2461

This theorem is referenced by:  rexlimddv  2599  nnsucuniel  6495  omp1eomlem  7092  ctmlemr  7106  mulgt0sr  7776  axpre-suploclemres  7899  cnegex  8134  receuap  8625  recapb  8627  rexanuz  10996  climcaucn  11358  fsumiun  11484  dvdsval2  11796  prmind2  12119  pcprmpw2  12331  pockthg  12354  dvdsrvald  13260  dvdsrd  13261  dvdsrex  13265  unitgrp  13283  tgcl  13534  neiint  13615  restopnb  13651  iscnp4  13688  blssexps  13899  blssex  13900  lgsne0  14409