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Theorem sbcnel12g 3957
Description: Distribute proper substitution through negated membership. (Contributed by Andrew Salmon, 18-Jun-2011.)
Assertion
Ref Expression
sbcnel12g (𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))

Proof of Theorem sbcnel12g
StepHypRef Expression
1 sbcng 3458 . 2 (𝐴𝑉 → ([𝐴 / 𝑥] ¬ 𝐵𝐶 ↔ ¬ [𝐴 / 𝑥]𝐵𝐶))
2 df-nel 2894 . . 3 (𝐵𝐶 ↔ ¬ 𝐵𝐶)
32sbcbii 3473 . 2 ([𝐴 / 𝑥]𝐵𝐶[𝐴 / 𝑥] ¬ 𝐵𝐶)
4 df-nel 2894 . . 3 (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶 ↔ ¬ 𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)
5 sbcel12 3955 . . 3 ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)
64, 5xchbinxr 325 . 2 (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶 ↔ ¬ [𝐴 / 𝑥]𝐵𝐶)
71, 3, 63bitr4g 303 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wcel 1987  wnel 2893  [wsbc 3417  csb 3514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-nel 2894  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-nul 3892
This theorem is referenced by:  rusbcALT  38122
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