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Theorem sbcng 3276
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 3238 . 2  |-  ( y  =  A  ->  ( [ y  /  x ]  -.  ph  <->  [. A  /  x ].  -.  ph ) )
2 dfsbcq2 3238 . . 3  |-  ( y  =  A  ->  ( [ y  /  x ] ph  <->  [. A  /  x ]. ph ) )
32notbid 295 . 2  |-  ( y  =  A  ->  ( -.  [ y  /  x ] ph  <->  -.  [. A  /  x ]. ph ) )
4 sbn 2190 . 2  |-  ( [ y  /  x ]  -.  ph  <->  -.  [ y  /  x ] ph )
51, 3, 4vtoclbg 3076 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 1790    e. wcel 1872   [.wsbc 3235
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-12 1909  ax-13 2058  ax-ext 2402
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2409  df-cleq 2415  df-clel 2418  df-v 3018  df-sbc 3236
This theorem is referenced by:  sbcn1  3281  sbcrext  3309  sbcnel12g  3740  sbcne12  3741  difopab  4921  bnj23  29469  bnj110  29614  bnj1204  29766  sbcni  32250  frege124d  36260  onfrALTlem5  36815  onfrALTlem5VD  37192
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