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  6649  omp1eomlem  7272  ctmlemr  7286  mulgt0sr  7976  axpre-suploclemres  8099  cnegex  8335  receuap  8827  recapb  8829  rexanuz  11514  climcaucn  11877  fsumiun  12003  dvdsval2  12316  nninfctlemfo  12576  prmind2  12657  pcprmpw2  12871  pockthg  12895  dvdsrvald  14072  dvdsrd  14073  dvdsrex  14077  unitgrp  14095  isnzr2  14163  znunit  14638  tgcl  14753  neiint  14834  restopnb  14870  iscnp4  14907  blssexps  15118  blssex  15119  lgsne0  15732  lgsquadlem1  15771