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Theorem sbcng 3372
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 3334 . 2  |-  ( y  =  A  ->  ( [ y  /  x ]  -.  ph  <->  [. A  /  x ].  -.  ph ) )
2 dfsbcq2 3334 . . 3  |-  ( y  =  A  ->  ( [ y  /  x ] ph  <->  [. A  /  x ]. ph ) )
32notbid 294 . 2  |-  ( y  =  A  ->  ( -.  [ y  /  x ] ph  <->  -.  [. A  /  x ]. ph ) )
4 sbn 2105 . 2  |-  ( [ y  /  x ]  -.  ph  <->  -.  [ y  /  x ] ph )
51, 3, 4vtoclbg 3172 1  |-  ( A  e.  V  ->  ( [. A  /  x ].  -.  ph  <->  -.  [. A  /  x ]. ph ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    = wceq 1379   [wsb 1711    e. wcel 1767   [.wsbc 3331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-v 3115  df-sbc 3332
This theorem is referenced by:  sbcn1  3379  sbcrextOLD  3413  sbcrext  3414  sbcnel12g  3826  sbcne12  3827  sbcne12gOLD  3828  difopab  5134  sbcni  30145  onfrALTlem5  32412  onfrALTlem5VD  32783  bnj23  32869  bnj110  33013  bnj1204  33165
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