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Theorem sbcn1 3375
Description: Move negation in and out of class substitution. One direction of sbcng 3368 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 3337 . 2  |-  ( [. A  /  x ].  -.  ph 
->  A  e.  _V )
2 sbcng 3368 . . 3  |-  ( A  e.  _V  ->  ( [. A  /  x ].  -.  ph  <->  -.  [. A  /  x ]. ph ) )
32biimpd 207 . 2  |-  ( A  e.  _V  ->  ( [. A  /  x ].  -.  ph  ->  -.  [. A  /  x ]. ph )
)
41, 3mpcom 36 1  |-  ( [. A  /  x ].  -.  ph 
->  -.  [. A  /  x ]. ph )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 1819   _Vcvv 3109   [.wsbc 3327
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-v 3111  df-sbc 3328
This theorem is referenced by: (None)
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