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Theorem sbcnel12g 2924
 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 df-nel 2341 . . . 4 (𝐵𝐶 ↔ ¬ 𝐵𝐶)
21sbcbii 2874 . . 3 ([𝐴 / 𝑥]𝐵𝐶[𝐴 / 𝑥] ¬ 𝐵𝐶)
32a1i 9 . 2 (𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝐶[𝐴 / 𝑥] ¬ 𝐵𝐶))
4 sbcng 2855 . 2 (𝐴𝑉 → ([𝐴 / 𝑥] ¬ 𝐵𝐶 ↔ ¬ [𝐴 / 𝑥]𝐵𝐶))
5 sbcel12g 2922 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
65notbid 625 . . 3 (𝐴𝑉 → (¬ [𝐴 / 𝑥]𝐵𝐶 ↔ ¬ 𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
7 df-nel 2341 . . 3 (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶 ↔ ¬ 𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)
86, 7syl6bbr 196 . 2 (𝐴𝑉 → (¬ [𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
93, 4, 83bitrd 212 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 103   ∈ wcel 1434   ∉ wnel 2340  [wsbc 2816  ⦋csb 2909 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064 This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-nel 2341  df-v 2604  df-sbc 2817  df-csb 2910 This theorem is referenced by: (None)
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