Theorem ralimdva 2617

Description: Deduction quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 22-May-1999.)

Hypothesis

Ref	Expression
ralimdva.1	|- ( ( ph /\ x e. A ) -> ( ps -> ch ) )

Assertion

Ref	Expression
ralimdva	|- ( ph -> ( A. x e. A ps -> A. x e. A ch ) )

Distinct variable group:   ph, x

Allowed substitution hints:   ps( x)   ch( x)   A( x)

Proof of Theorem ralimdva

Step	Hyp	Ref	Expression
1		nfv 1581	. 2 |- F/ x ph
2		ralimdva.1	. 2 |- ( ( ph /\ x e. A ) -> ( ps -> ch ) )
3	1, 2	ralimdaa 2616	1 |- ( ph -> ( A. x e. A ps -> A. x e. A ch ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2209   A.wral 2528

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 1500  ax-gen 1502  ax-4 1563  ax-17 1579

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-ral 2533

This theorem is referenced by:  ralimdv  2618  ralimdvva  2619  f1mpt  5967  isores3  6011  caofrss  6324  caoftrn  6325  tfrlemibxssdm  6588  tfr1onlembxssdm  6604  tfrcllembxssdm  6617  tfrcl  6625  infidc  7238  exmidomniim  7471  exmidontri2or  7592  caucvgsrlemoffcau  8155  caucvgsrlemoffres  8157  indstr  9972  caucvgre  11725  rexuz3  11734  resqrexlemgt0  11764  resqrexlemglsq  11766  cau3lem  11858  rexanre  11964  rexico  11965  2clim  12045  climcn1  12052  climcn2  12053  subcn2  12055  climsqz  12079  climsqz2  12080  climcvg1nlem  12093  fprodsplitdc  12341  bezoutlemaz  12758  bezoutlembz  12759  bezoutlembi  12760  pcfac  13107  pockthg  13114  infpnlem1  13116  isgrpinv  13836  dfgrp3me  13882  issubg4m  13973  mplsubgfileminv  15014  cncnp  15254  txlm  15303  metequiv2  15520  metcnpi3  15541  rescncf  15605  cncfco  15615  suplociccreex  15648  limcresi  15690  cnplimcim  15691  cnplimclemr  15693  cnlimcim  15695  limccnpcntop  15699  limccoap  15702  2sqlem6  16153  wlkvtxiedg  16500  wlkvtxiedgg  16501  upgrwlkvtxedg  16519  uspgr2wlkeq  16520  clwwlkccatlem  16555  bj-charfunbi  16751  nninffeq  16968  tridceq  17011