Theorem rexlimdvaa 2651

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 2650	1 |- ( ph -> ( E. x e. A ps -> ch ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2202   E.wrex 2511

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 1495  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-4 1558  ax-17 1574  ax-ial 1582  ax-i5r 1583

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-ral 2515  df-rex 2516

This theorem is referenced by:  rexlimddv  2655  nnsucuniel  6663  omp1eomlem  7293  ctmlemr  7307  mulgt0sr  7998  axpre-suploclemres  8121  cnegex  8357  receuap  8849  recapb  8851  rexanuz  11549  climcaucn  11912  fsumiun  12039  dvdsval2  12352  nninfctlemfo  12612  prmind2  12693  pcprmpw2  12907  pockthg  12931  dvdsrvald  14109  dvdsrd  14110  dvdsrex  14114  unitgrp  14132  isnzr2  14200  znunit  14675  tgcl  14790  neiint  14871  restopnb  14907  iscnp4  14944  blssexps  15155  blssex  15156  lgsne0  15769  lgsquadlem1  15808