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Theorem sbcn1 3772
Description: Move negation in and out of class substitution. One direction of sbcng 3767 that holds for proper classes. (Contributed by NM, 17-Aug-2018.)
Assertion
Ref Expression
sbcn1 ([𝐴 / 𝑥] ¬ 𝜑 → ¬ [𝐴 / 𝑥]𝜑)

Proof of Theorem sbcn1
StepHypRef Expression
1 sbcex 3727 . 2 ([𝐴 / 𝑥] ¬ 𝜑𝐴 ∈ V)
2 sbcng 3767 . . 3 (𝐴 ∈ V → ([𝐴 / 𝑥] ¬ 𝜑 ↔ ¬ [𝐴 / 𝑥]𝜑))
32biimpd 228 . 2 (𝐴 ∈ V → ([𝐴 / 𝑥] ¬ 𝜑 → ¬ [𝐴 / 𝑥]𝜑))
41, 3mpcom 38 1 ([𝐴 / 𝑥] ¬ 𝜑 → ¬ [𝐴 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 2106  Vcvv 3431  [wsbc 3717
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3433  df-sbc 3718
This theorem is referenced by: (None)
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