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Theorem sbcng 2921
Description: Move negation in and out of class substitution. (Contributed by NM, 16-Jan-2004.)
Assertion
Ref Expression
sbcng  |-  ( A  e.  V  ->  ( [. A  /  x ].  -.  ph  <->  -.  [. A  /  x ]. ph ) )

Proof of Theorem sbcng
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 2885 . 2  |-  ( y  =  A  ->  ( [ y  /  x ]  -.  ph  <->  [. A  /  x ].  -.  ph ) )
2 dfsbcq2 2885 . . 3  |-  ( y  =  A  ->  ( [ y  /  x ] ph  <->  [. A  /  x ]. ph ) )
32notbid 641 . 2  |-  ( y  =  A  ->  ( -.  [ y  /  x ] ph  <->  -.  [. A  /  x ]. ph ) )
4 sbn 1903 . 2  |-  ( [ y  /  x ]  -.  ph  <->  -.  [ y  /  x ] ph )
51, 3, 4vtoclbg 2721 1  |-  ( A  e.  V  ->  ( [. A  /  x ].  -.  ph  <->  -.  [. A  /  x ]. ph ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 104    = wceq 1316    e. wcel 1465   [wsb 1720   [.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-in1 588  ax-in2 589  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-fal 1322  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:  sbcn1  2928  sbcnel12g  2990  sbcne12g  2991  difopab  4642  zsupcllemstep  11565
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