Theorem rexlimdvaa 2669

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 2668	1 |- ( ph -> ( E. x e. A ps -> ch ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   e. wcel 2209   E.wrex 2529

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587  ax-i5r 1588

This proof depends on definitions:  df-bi 117  df-nf 1514  df-ral 2533  df-rex 2534

This theorem is used by:  rexlimddv  2673  nnsucuniel  6768  omp1eomlem  7434  ctmlemr  7448  mulgt0sr  8145  axpre-suploclemres  8268  cnegex  8504  receuap  8999  recapb  9001  rexanuz  11754  climcaucn  12117  fsumiun  12244  dvdsval2  12557  nninfctlemfo  12817  prmind2  12898  pcprmpw2  13112  pockthg  13136  dvdsrvald  14400  dvdsrd  14401  dvdsrex  14405  unitgrp  14423  isnzr2  14491  znunit  14994  tgcl  15165  neiint  15246  restopnb  15282  iscnp4  15319  blssexps  15530  blssex  15531  lgsne0  16157  lgsquadlem1  16196