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Theorem vtocl 2667
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 1464 . 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 2666 1  |-  ps
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1287    e. wcel 1436   _Vcvv 2615
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-5 1379  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-v 2617
This theorem is referenced by:  vtoclb  2670  zfauscl  3934  bnd2  3983  uniex  4238  ordtriexmid  4311  onsucsssucexmid  4316  regexmid  4324  ordsoexmid  4351  onintexmid  4361  reg3exmid  4368  nnregexmid  4407  acexmidlemv  5611  caovcan  5766  findcard2  6557  findcard2s  6558  inffiexmid  6574  bj-uniex  11246  bj-omtrans  11289
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