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Theorem vtocl 2735
Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 30-Aug-1993.)
Hypotheses
Ref Expression
vtocl.1  |-  A  e. 
_V
vtocl.2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
vtocl.3  |-  ph
Assertion
Ref Expression
vtocl  |-  ps
Distinct variable groups:    x, A    ps, x
Allowed substitution hint:    ph( x)

Proof of Theorem vtocl
StepHypRef Expression
1 nfv 1508 . 2  |-  F/ x ps
2 vtocl.1 . 2  |-  A  e. 
_V
3 vtocl.2 . 2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
4 vtocl.3 . 2  |-  ph
51, 2, 3, 4vtoclf 2734 1  |-  ps
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1331    e. wcel 1480   _Vcvv 2681
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-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-v 2683
This theorem is referenced by:  vtoclb  2738  zfauscl  4043  bnd2  4092  uniex  4354  ordtriexmid  4432  onsucsssucexmid  4437  regexmid  4445  ordsoexmid  4472  onintexmid  4482  reg3exmid  4489  nnregexmid  4529  acexmidlemv  5765  caovcan  5928  findcard2  6776  findcard2s  6777  inffiexmid  6793  sup3exmid  8708  bj-uniex  13104  bj-omtrans  13143
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