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Theorem sbcni 37812
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 3827 . . 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 333 1 ([𝐴 / 𝑥] ¬ 𝜑 ↔ ¬ 𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wcel 2099  Vcvv 3462  [wsbc 3776
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-12 2167  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-sbc 3777
This theorem is referenced by: (None)
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