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Theorem sbc7 2908
Description: An equivalence for class substitution in the spirit of df-clab 2104. Note that  x and  A don't have to be distinct. (Contributed by NM, 18-Nov-2008.) (Revised by Mario Carneiro, 13-Oct-2016.)
Assertion
Ref Expression
sbc7  |-  ( [. A  /  x ]. ph  <->  E. y
( y  =  A  /\  [. y  /  x ]. ph ) )
Distinct variable groups:    y, A    ph, y    x, y
Allowed substitution hints:    ph( x)    A( x)

Proof of Theorem sbc7
StepHypRef Expression
1 sbcco 2903 . 2  |-  ( [. A  /  y ]. [. y  /  x ]. ph  <->  [. A  /  x ]. ph )
2 sbc5 2905 . 2  |-  ( [. A  /  y ]. [. y  /  x ]. ph  <->  E. y
( y  =  A  /\  [. y  /  x ]. ph ) )
31, 2bitr3i 185 1  |-  ( [. A  /  x ]. ph  <->  E. y
( y  =  A  /\  [. y  /  x ]. ph ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    = wceq 1316   E.wex 1453   [.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: (None)
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