Theorem rexlimdvaa 2652

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 2651	1 |- ( ph -> ( E. x e. A ps -> ch ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2202   E.wrex 2512

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 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-17 1575  ax-ial 1583  ax-i5r 1584

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-ral 2516  df-rex 2517

This theorem is referenced by:  rexlimddv  2656  nnsucuniel  6706  omp1eomlem  7336  ctmlemr  7350  mulgt0sr  8041  axpre-suploclemres  8164  cnegex  8399  receuap  8891  recapb  8893  rexanuz  11611  climcaucn  11974  fsumiun  12101  dvdsval2  12414  nninfctlemfo  12674  prmind2  12755  pcprmpw2  12969  pockthg  12993  dvdsrvald  14171  dvdsrd  14172  dvdsrex  14176  unitgrp  14194  isnzr2  14262  znunit  14738  tgcl  14858  neiint  14939  restopnb  14975  iscnp4  15012  blssexps  15223  blssex  15224  lgsne0  15840  lgsquadlem1  15879