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Theorem vtoclga 2636
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 2194 . 2  |-  F/_ x A
2 nfv 1437 . 2  |-  F/ x ps
3 vtoclga.1 . 2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
4 vtoclga.2 . 2  |-  ( x  e.  B  ->  ph )
51, 2, 3, 4vtoclgaf 2635 1  |-  ( A  e.  B  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 102    = wceq 1259    e. wcel 1409
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576
This theorem is referenced by:  vtoclri  2645  ssuni  3629  ordtriexmid  4274  onsucsssucexmid  4279  tfis3  4336  fvmpt3  5278  fvmptssdm  5282  fnressn  5376  fressnfv  5377  caovord  5699  caovimo  5721  tfrlem1  5953  freccl  6023  nnacl  6089  nnmcl  6090  nnacom  6093  nnaass  6094  nndi  6095  nnmass  6096  nnmsucr  6097  nnmcom  6098  nnsucsssuc  6101  nntri3or  6102  nnaordi  6111  nnaword  6114  nnmordi  6119  nnaordex  6130  findcard  6375  findcard2  6376  findcard2s  6377  indpi  6497  prarloclem3  6652  uzind4s2  8629  cnref1o  8679  frec2uzrdg  9358  expcl2lemap  9431  climub  10094  climserile  10095  ialginv  10248  ialgcvg  10249  ialgcvga  10252  ialgfx  10253
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