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Theorem sbc6 3130
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 3129 . 2  |-  ( A  e.  _V  ->  ( [. A  /  x ]. ph  <->  A. x ( x  =  A  ->  ph )
) )
31, 2ax-mp 8 1  |-  ( [. A  /  x ]. ph  <->  A. x
( x  =  A  ->  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1546    = wceq 1649    e. wcel 1717   _Vcvv 2899   [.wsbc 3104
This theorem is referenced by:  intab  4022  2sbc6g  27284
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-v 2901  df-sbc 3105
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