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  11566  climcaucn  11929  fsumiun  12056  dvdsval2  12369  nninfctlemfo  12629  prmind2  12710  pcprmpw2  12924  pockthg  12948  dvdsrvald  14126  dvdsrd  14127  dvdsrex  14131  unitgrp  14149  isnzr2  14217  znunit  14692  tgcl  14807  neiint  14888  restopnb  14924  iscnp4  14961  blssexps  15172  blssex  15173  lgsne0  15786  lgsquadlem1  15825