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Theorem vtoclga 2685
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 2228 . 2  |-  F/_ x A
2 nfv 1466 . 2  |-  F/ x ps
3 vtoclga.1 . 2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
4 vtoclga.2 . 2  |-  ( x  e.  B  ->  ph )
51, 2, 3, 4vtoclgaf 2684 1  |-  ( A  e.  B  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1289    e. wcel 1438
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621
This theorem is referenced by:  vtoclri  2694  ssuni  3675  ordtriexmid  4338  onsucsssucexmid  4343  tfis3  4401  fvmpt3  5383  fvmptssdm  5387  fnressn  5483  fressnfv  5484  caovord  5816  caovimo  5838  tfrlem1  6073  nnacl  6241  nnmcl  6242  nnacom  6245  nnaass  6246  nndi  6247  nnmass  6248  nnmsucr  6249  nnmcom  6250  nnsucsssuc  6253  nntri3or  6254  nnaordi  6267  nnaword  6270  nnmordi  6275  nnaordex  6286  findcard  6604  findcard2  6605  findcard2s  6606  exmidomni  6798  indpi  6901  prarloclem3  7056  uzind4s2  9079  cnref1o  9133  frec2uzrdg  9816  expcl2lemap  9967  iseqcoll  10247  climub  10733  climserle  10734  fisumcvg  10766  fsum3cvg  10767  isummolem2a  10771  ialginv  11307  ialgcvg  11308  ialgcvga  11311  ialgfx  11312  prmind2  11380
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