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Theorem sbcng 3346
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 3308 . 2  |-  ( y  =  A  ->  ( [ y  /  x ]  -.  ph  <->  [. A  /  x ].  -.  ph ) )
2 dfsbcq2 3308 . . 3  |-  ( y  =  A  ->  ( [ y  /  x ] ph  <->  [. A  /  x ]. ph ) )
32notbid 295 . 2  |-  ( y  =  A  ->  ( -.  [ y  /  x ] ph  <->  -.  [. A  /  x ]. ph ) )
4 sbn 2186 . 2  |-  ( [ y  /  x ]  -.  ph  <->  -.  [ y  /  x ] ph )
51, 3, 4vtoclbg 3146 1  |-  ( A  e.  V  ->  ( [. A  /  x ].  -.  ph  <->  -.  [. A  /  x ]. ph ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    = wceq 1437   [wsb 1789    e. wcel 1870   [.wsbc 3305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-v 3089  df-sbc 3306
This theorem is referenced by:  sbcn1  3351  sbcrext  3379  sbcnel12g  3808  sbcne12  3809  difopab  4986  bnj23  29312  bnj110  29457  bnj1204  29609  sbcni  32053  frege124d  35992  onfrALTlem5  36545  onfrALTlem5VD  36922
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