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Theorem sbcn1 2908
Description: Move negation in and out of class substitution. One direction of sbcng 2901 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 2870 . 2  |-  ( [. A  /  x ].  -.  ph 
->  A  e.  _V )
2 sbcng 2901 . . 3  |-  ( A  e.  _V  ->  ( [. A  /  x ].  -.  ph  <->  -.  [. A  /  x ]. ph ) )
32biimpd 143 . 2  |-  ( A  e.  _V  ->  ( [. A  /  x ].  -.  ph  ->  -.  [. A  /  x ]. ph )
)
41, 3mpcom 36 1  |-  ( [. A  /  x ].  -.  ph 
->  -.  [. A  /  x ]. ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 1448   _Vcvv 2641   [.wsbc 2862
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643  df-sbc 2863
This theorem is referenced by: (None)
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