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  6773  omp1eomlem  7440  ctmlemr  7454  mulgt0sr  8151  axpre-suploclemres  8274  cnegex  8511  receuap  9007  recapb  9009  rexanuz  11775  fiidxsupcl  12017  climcaucn  12141  fsumiun  12268  dvdsval2  12581  nninfctlemfo  12841  prmind2  12922  nn0sqdcq  13012  sqrtrirr  13013  pcprmpw2  13140  pockthg  13164  dvdsrvald  14508  dvdsrd  14509  dvdsrex  14513  unitgrp  14531  isnzr2  14599  znunit  15102  tgcl  15307  neiint  15388  restopnb  15424  iscnp4  15461  blssexps  15672  blssex  15673  lgsne0  16379  lgsquadlem1  16418