Theorem rexlimdvaa 2626

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 2625	1 |- ( ph -> ( E. x e. A ps -> ch ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2178   E.wrex 2487

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 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-4 1534  ax-17 1550  ax-ial 1558  ax-i5r 1559

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-ral 2491  df-rex 2492

This theorem is referenced by:  rexlimddv  2630  nnsucuniel  6604  omp1eomlem  7222  ctmlemr  7236  mulgt0sr  7926  axpre-suploclemres  8049  cnegex  8285  receuap  8777  recapb  8779  rexanuz  11414  climcaucn  11777  fsumiun  11903  dvdsval2  12216  nninfctlemfo  12476  prmind2  12557  pcprmpw2  12771  pockthg  12795  dvdsrvald  13970  dvdsrd  13971  dvdsrex  13975  unitgrp  13993  isnzr2  14061  znunit  14536  tgcl  14651  neiint  14732  restopnb  14768  iscnp4  14805  blssexps  15016  blssex  15017  lgsne0  15630  lgsquadlem1  15669