Users' Mathboxes Mathbox for Giovanni Mascellani < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sbcni Structured version   Visualization version   GIF version

Theorem sbcni 35270
Description: Move class substitution inside a negation, in inference form. (Contributed by Giovanni Mascellani, 27-May-2019.)
Hypotheses
Ref Expression
sbcni.1 𝐴 ∈ V
sbcni.2 ([𝐴 / 𝑥]𝜑𝜓)
Assertion
Ref Expression
sbcni ([𝐴 / 𝑥] ¬ 𝜑 ↔ ¬ 𝜓)

Proof of Theorem sbcni
StepHypRef Expression
1 sbcni.1 . . 3 𝐴 ∈ V
2 sbcng 3816 . . 3 (𝐴 ∈ V → ([𝐴 / 𝑥] ¬ 𝜑 ↔ ¬ [𝐴 / 𝑥]𝜑))
31, 2ax-mp 5 . 2 ([𝐴 / 𝑥] ¬ 𝜑 ↔ ¬ [𝐴 / 𝑥]𝜑)
4 sbcni.2 . 2 ([𝐴 / 𝑥]𝜑𝜓)
53, 4xchbinx 335 1 ([𝐴 / 𝑥] ¬ 𝜑 ↔ ¬ 𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 207  wcel 2105  Vcvv 3492  [wsbc 3769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-sbc 3770
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator