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Theorem sbc6g 2906
Description: An equivalence for class substitution. (Contributed by NM, 11-Oct-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
Assertion
Ref Expression
sbc6g  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  A. x ( x  =  A  ->  ph )
) )
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    V( x)

Proof of Theorem sbc6g
StepHypRef Expression
1 nfe1 1457 . . 3  |-  F/ x E. x ( x  =  A  /\  ph )
2 ceqex 2786 . . 3  |-  ( x  =  A  ->  ( ph 
<->  E. x ( x  =  A  /\  ph ) ) )
31, 2ceqsalg 2688 . 2  |-  ( A  e.  V  ->  ( A. x ( x  =  A  ->  ph )  <->  E. x
( x  =  A  /\  ph ) ) )
4 sbc5 2905 . 2  |-  ( [. A  /  x ]. ph  <->  E. x
( x  =  A  /\  ph ) )
53, 4syl6rbbr 198 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  A. x ( x  =  A  ->  ph )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   A.wal 1314    = wceq 1316   E.wex 1453    e. wcel 1465   [.wsbc 2882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-sbc 2883
This theorem is referenced by:  sbc6  2907  sbciegft  2911  ralsnsg  3531  ralsns  3532  fz1sbc  9844
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