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Theorem sbc6 2863
Description: An equivalence for class substitution. (Contributed by NM, 23-Aug-1993.) (Proof shortened by Eric Schmidt, 17-Jan-2007.)
Hypothesis
Ref Expression
sbc6.1  |-  A  e. 
_V
Assertion
Ref Expression
sbc6  |-  ( [. A  /  x ]. ph  <->  A. x
( x  =  A  ->  ph ) )
Distinct variable group:    x, A
Allowed substitution hint:    ph( x)

Proof of Theorem sbc6
StepHypRef Expression
1 sbc6.1 . 2  |-  A  e. 
_V
2 sbc6g 2862 . 2  |-  ( A  e.  _V  ->  ( [. A  /  x ]. ph  <->  A. x ( x  =  A  ->  ph )
) )
31, 2ax-mp 7 1  |-  ( [. A  /  x ]. ph  <->  A. x
( x  =  A  ->  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wal 1287    = wceq 1289    e. wcel 1438   _Vcvv 2619   [.wsbc 2838
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  df-sbc 2839
This theorem is referenced by:  intab  3712
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