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Theorem sbcni 32257
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 3337 . . 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 311 1  |-  ( [. A  /  x ].  -.  ph  <->  -. 
ps )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 187    e. wcel 1867   _Vcvv 3078   [.wsbc 3296
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-10 1886  ax-12 1904  ax-13 2052  ax-ext 2398
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2406  df-cleq 2412  df-clel 2415  df-v 3080  df-sbc 3297
This theorem is referenced by: (None)
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