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Theorem sbcnel12g 4413
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 3835 . 2 (𝐴𝑉 → ([𝐴 / 𝑥] ¬ 𝐵𝐶 ↔ ¬ [𝐴 / 𝑥]𝐵𝐶))
2 df-nel 3046 . . 3 (𝐵𝐶 ↔ ¬ 𝐵𝐶)
32sbcbii 3845 . 2 ([𝐴 / 𝑥]𝐵𝐶[𝐴 / 𝑥] ¬ 𝐵𝐶)
4 df-nel 3046 . . 3 (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶 ↔ ¬ 𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)
5 sbcel12 4410 . . 3 ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)
64, 5xchbinxr 335 . 2 (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶 ↔ ¬ [𝐴 / 𝑥]𝐵𝐶)
71, 3, 63bitr4g 314 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wcel 2107  wnel 3045  [wsbc 3787  csb 3898
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-nel 3046  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-nul 4333
This theorem is referenced by:  rusbcALT  44463
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