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Theorem sbcnel12g 4129
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 3628 . 2 (𝐴𝑉 → ([𝐴 / 𝑥] ¬ 𝐵𝐶 ↔ ¬ [𝐴 / 𝑥]𝐵𝐶))
2 df-nel 3047 . . 3 (𝐵𝐶 ↔ ¬ 𝐵𝐶)
32sbcbii 3643 . 2 ([𝐴 / 𝑥]𝐵𝐶[𝐴 / 𝑥] ¬ 𝐵𝐶)
4 df-nel 3047 . . 3 (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶 ↔ ¬ 𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)
5 sbcel12 4127 . . 3 ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)
64, 5xchbinxr 324 . 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 2145  wnel 3046  [wsbc 3587  csb 3682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-nel 3047  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-nul 4064
This theorem is referenced by:  rusbcALT  39166
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