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Theorem sbcni 34546
 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 3694 . . 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 326 1 ([𝐴 / 𝑥] ¬ 𝜑 ↔ ¬ 𝜓)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 198   ∈ wcel 2107  Vcvv 3398  [wsbc 3652 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-12 2163  ax-13 2334  ax-ext 2754 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-v 3400  df-sbc 3653 This theorem is referenced by: (None)
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