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Theorem sbcni 36248
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 3769 . . 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 2109  Vcvv 3430  [wsbc 3719
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-12 2174  ax-ext 2710
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1544  df-ex 1786  df-nf 1790  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-sbc 3720
This theorem is referenced by: (None)
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