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Theorem vtoclga 2674
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 2223 . 2  |-  F/_ x A
2 nfv 1462 . 2  |-  F/ x ps
3 vtoclga.1 . 2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
4 vtoclga.2 . 2  |-  ( x  e.  B  ->  ph )
51, 2, 3, 4vtoclgaf 2673 1  |-  ( A  e.  B  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1285    e. wcel 1434
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2613
This theorem is referenced by:  vtoclri  2683  ssuni  3644  ordtriexmid  4294  onsucsssucexmid  4299  tfis3  4356  fvmpt3  5305  fvmptssdm  5309  fnressn  5403  fressnfv  5404  caovord  5725  caovimo  5747  tfrlem1  5979  nnacl  6146  nnmcl  6147  nnacom  6150  nnaass  6151  nndi  6152  nnmass  6153  nnmsucr  6154  nnmcom  6155  nnsucsssuc  6158  nntri3or  6159  nnaordi  6170  nnaword  6173  nnmordi  6178  nnaordex  6189  findcard  6446  findcard2  6447  findcard2s  6448  exmidomni  6583  indpi  6671  prarloclem3  6826  uzind4s2  8837  cnref1o  8891  frec2uzrdg  9568  expcl2lemap  9662  climub  10408  climserile  10409  fisumcvg  10426  ialginv  10661  ialgcvg  10662  ialgcvga  10665  ialgfx  10666  prmind2  10734
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