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  6662  omp1eomlem  7292  ctmlemr  7306  mulgt0sr  7997  axpre-suploclemres  8120  cnegex  8356  receuap  8848  recapb  8850  rexanuz  11548  climcaucn  11911  fsumiun  12037  dvdsval2  12350  nninfctlemfo  12610  prmind2  12691  pcprmpw2  12905  pockthg  12929  dvdsrvald  14106  dvdsrd  14107  dvdsrex  14111  unitgrp  14129  isnzr2  14197  znunit  14672  tgcl  14787  neiint  14868  restopnb  14904  iscnp4  14941  blssexps  15152  blssex  15153  lgsne0  15766  lgsquadlem1  15805