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Theorem vtoclga 2778
Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 20-Aug-1995.)
Hypotheses
Ref Expression
vtoclga.1  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
vtoclga.2  |-  ( x  e.  B  ->  ph )
Assertion
Ref Expression
vtoclga  |-  ( A  e.  B  ->  ps )
Distinct variable groups:    x, A    x, B    ps, x
Allowed substitution hint:    ph( x)

Proof of Theorem vtoclga
StepHypRef Expression
1 nfcv 2299 . 2  |-  F/_ x A
2 nfv 1508 . 2  |-  F/ x ps
3 vtoclga.1 . 2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
4 vtoclga.2 . 2  |-  ( x  e.  B  ->  ph )
51, 2, 3, 4vtoclgaf 2777 1  |-  ( A  e.  B  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1335    e. wcel 2128
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714
This theorem is referenced by:  vtoclri  2787  ssuni  3794  ordtriexmid  4480  onsucsssucexmid  4486  tfis3  4545  fvmpt3  5547  fvmptssdm  5552  fnressn  5653  fressnfv  5654  caovord  5992  caovimo  6014  tfrlem1  6255  nnacl  6427  nnmcl  6428  nnacom  6431  nnaass  6432  nndi  6433  nnmass  6434  nnmsucr  6435  nnmcom  6436  nnsucsssuc  6439  nntri3or  6440  nnaordi  6455  nnaword  6458  nnmordi  6463  nnaordex  6474  ixpfn  6649  findcard  6833  findcard2  6834  findcard2s  6835  exmidomni  7085  indpi  7262  prarloclem3  7417  uzind4s2  9502  cnref1o  9559  frec2uzrdg  10308  expcl2lemap  10431  seq3coll  10713  climub  11241  climserle  11242  fsum3cvg  11275  summodclem2a  11278  prodfap0  11442  prodfrecap  11443  fproddccvg  11469  alginv  11924  algcvg  11925  algcvga  11928  algfx  11929  prmind2  11997
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