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  7435  ctmlemr  7449  mulgt0sr  8146  axpre-suploclemres  8269  cnegex  8506  receuap  9002  recapb  9004  rexanuz  11770  fiidxsupcl  12012  climcaucn  12136  fsumiun  12263  dvdsval2  12576  nninfctlemfo  12836  prmind2  12917  nn0sqdcq  13007  sqrtrirr  13008  pcprmpw2  13135  pockthg  13159  dvdsrvald  14484  dvdsrd  14485  dvdsrex  14489  unitgrp  14507  isnzr2  14575  znunit  15078  tgcl  15256  neiint  15337  restopnb  15373  iscnp4  15410  blssexps  15621  blssex  15622  lgsne0  16323  lgsquadlem1  16362