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Theorem vtocl 2815
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 1539 . 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 2814 1  |-  ps
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105    = wceq 1364    e. wcel 2164   _Vcvv 2760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-ext 2175
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-v 2762
This theorem is referenced by:  vtoclb  2818  zfauscl  4150  bnd2  4203  uniex  4469  ordtriexmid  4554  onsucsssucexmid  4560  regexmid  4568  ordsoexmid  4595  onintexmid  4606  reg3exmid  4613  nnregexmid  4654  acexmidlemv  5917  caovcan  6085  findcard2  6947  findcard2s  6948  inffiexmid  6964  sup3exmid  8978  bj-uniex  15479  bj-omtrans  15518
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