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Theorem sbcn1 3313
Description: Move negation in and out of class substitution. One direction of sbcng 3308 that holds for proper classes. (Contributed by NM, 17-Aug-2018.)
Assertion
Ref Expression
sbcn1  |-  ( [. A  /  x ].  -.  ph 
->  -.  [. A  /  x ]. ph )

Proof of Theorem sbcn1
StepHypRef Expression
1 sbcex 3277 . 2  |-  ( [. A  /  x ].  -.  ph 
->  A  e.  _V )
2 sbcng 3308 . . 3  |-  ( A  e.  _V  ->  ( [. A  /  x ].  -.  ph  <->  -.  [. A  /  x ]. ph ) )
32biimpd 211 . 2  |-  ( A  e.  _V  ->  ( [. A  /  x ].  -.  ph  ->  -.  [. A  /  x ]. ph )
)
41, 3mpcom 37 1  |-  ( [. A  /  x ].  -.  ph 
->  -.  [. A  /  x ]. ph )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 1887   _Vcvv 3045   [.wsbc 3267
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-12 1933  ax-13 2091  ax-ext 2431
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-v 3047  df-sbc 3268
This theorem is referenced by: (None)
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