Theorem rexlimdvaa 2661

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 2660	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:  rexlimddv  2665  nnsucuniel  6728  omp1eomlem  7385  ctmlemr  7399  mulgt0sr  8093  axpre-suploclemres  8216  cnegex  8451  receuap  8943  recapb  8945  rexanuz  11673  climcaucn  12036  fsumiun  12163  dvdsval2  12476  nninfctlemfo  12736  prmind2  12817  pcprmpw2  13031  pockthg  13055  dvdsrvald  14238  dvdsrd  14239  dvdsrex  14243  unitgrp  14261  isnzr2  14329  znunit  14807  tgcl  14929  neiint  15010  restopnb  15046  iscnp4  15083  blssexps  15294  blssex  15295  lgsne0  15911  lgsquadlem1  15950