Theorem rexlimdvaa 2649

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 2648	1 |- ( ph -> ( E. x e. A ps -> ch ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2200   E.wrex 2509

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 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-ial 1580  ax-i5r 1581

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-ral 2513  df-rex 2514

This theorem is referenced by:  rexlimddv  2653  nnsucuniel  6658  omp1eomlem  7284  ctmlemr  7298  mulgt0sr  7988  axpre-suploclemres  8111  cnegex  8347  receuap  8839  recapb  8841  rexanuz  11539  climcaucn  11902  fsumiun  12028  dvdsval2  12341  nninfctlemfo  12601  prmind2  12682  pcprmpw2  12896  pockthg  12920  dvdsrvald  14097  dvdsrd  14098  dvdsrex  14102  unitgrp  14120  isnzr2  14188  znunit  14663  tgcl  14778  neiint  14859  restopnb  14895  iscnp4  14932  blssexps  15143  blssex  15144  lgsne0  15757  lgsquadlem1  15796