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  6639  omp1eomlem  7257  ctmlemr  7271  mulgt0sr  7961  axpre-suploclemres  8084  cnegex  8320  receuap  8812  recapb  8814  rexanuz  11494  climcaucn  11857  fsumiun  11983  dvdsval2  12296  nninfctlemfo  12556  prmind2  12637  pcprmpw2  12851  pockthg  12875  dvdsrvald  14051  dvdsrd  14052  dvdsrex  14056  unitgrp  14074  isnzr2  14142  znunit  14617  tgcl  14732  neiint  14813  restopnb  14849  iscnp4  14886  blssexps  15097  blssex  15098  lgsne0  15711  lgsquadlem1  15750