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Theorem sbc6 3017
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 3016 . 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 176   A.wal 1527    = wceq 1623    e. wcel 1684   _Vcvv 2788   [.wsbc 2991
This theorem is referenced by:  intab  3892  2sbc6g  27615
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-sbc 2992
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