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Theorem sbcni 32342
Description: Move class substitution inside a negation, in inference form. (Contributed by Giovanni Mascellani, 27-May-2019.)
Hypotheses
Ref Expression
sbcni.1  |-  A  e. 
_V
sbcni.2  |-  ( [. A  /  x ]. ph  <->  ps )
Assertion
Ref Expression
sbcni  |-  ( [. A  /  x ].  -.  ph  <->  -. 
ps )

Proof of Theorem sbcni
StepHypRef Expression
1 sbcni.1 . . 3  |-  A  e. 
_V
2 sbcng 3307 . . 3  |-  ( A  e.  _V  ->  ( [. A  /  x ].  -.  ph  <->  -.  [. A  /  x ]. ph ) )
31, 2ax-mp 5 . 2  |-  ( [. A  /  x ].  -.  ph  <->  -. 
[. A  /  x ]. ph )
4 sbcni.2 . 2  |-  ( [. A  /  x ]. ph  <->  ps )
53, 4xchbinx 312 1  |-  ( [. A  /  x ].  -.  ph  <->  -. 
ps )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 188    e. wcel 1886   _Vcvv 3044   [.wsbc 3266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-10 1914  ax-12 1932  ax-13 2090  ax-ext 2430
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-clab 2437  df-cleq 2443  df-clel 2446  df-v 3046  df-sbc 3267
This theorem is referenced by: (None)
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