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Theorem sbc6 3179
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 3178 . 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 1549    = wceq 1652    e. wcel 1725   _Vcvv 2948   [.wsbc 3153
This theorem is referenced by:  intab  4072  2sbc6g  27573
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-sbc 3154
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