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  6772  omp1eomlem  7440  ctmlemr  7454  mulgt0sr  8151  axpre-suploclemres  8274  cnegex  8511  receuap  9007  recapb  9009  rexanuz  11776  fiidxsupcl  12018  climcaucn  12142  fsumiun  12269  dvdsval2  12582  nninfctlemfo  12842  prmind2  12923  nn0sqdcq  13013  sqrtrirr  13014  pcprmpw2  13141  pockthg  13165  dvdsrvald  14509  dvdsrd  14510  dvdsrex  14514  unitgrp  14532  isnzr2  14600  znunit  15103  tgcl  15290  neiint  15371  restopnb  15407  iscnp4  15444  blssexps  15655  blssex  15656  lgsne0  16362  lgsquadlem1  16401