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Theorem sbc6 2850
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 2849 . 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 1283    = wceq 1285    e. wcel 1434   _Vcvv 2610   [.wsbc 2825
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-sbc 2826
This theorem is referenced by:  intab  3686
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