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Theorem sbcne12g 3075
Description: Distribute proper substitution through an inequality. (Contributed by Andrew Salmon, 18-Jun-2011.)
Assertion
Ref Expression
sbcne12g (𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))

Proof of Theorem sbcne12g
StepHypRef Expression
1 sbceqg 3073 . . 3 (𝐴𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶))
21notbid 667 . 2 (𝐴𝑉 → (¬ [𝐴 / 𝑥]𝐵 = 𝐶 ↔ ¬ 𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶))
3 df-ne 2348 . . . . 5 (𝐵𝐶 ↔ ¬ 𝐵 = 𝐶)
43sbcbii 3022 . . . 4 ([𝐴 / 𝑥]𝐵𝐶[𝐴 / 𝑥] ¬ 𝐵 = 𝐶)
5 sbcng 3003 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥] ¬ 𝐵 = 𝐶 ↔ ¬ [𝐴 / 𝑥]𝐵 = 𝐶))
64, 5bitrid 192 . . 3 (𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝐶 ↔ ¬ [𝐴 / 𝑥]𝐵 = 𝐶))
7 df-ne 2348 . . . 4 (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶 ↔ ¬ 𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶)
87a1i 9 . . 3 (𝐴𝑉 → (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶 ↔ ¬ 𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶))
96, 8bibi12d 235 . 2 (𝐴𝑉 → (([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶) ↔ (¬ [𝐴 / 𝑥]𝐵 = 𝐶 ↔ ¬ 𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶)))
102, 9mpbird 167 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 105   = wceq 1353  wcel 2148  wne 2347  [wsbc 2962  csb 3057
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-v 2739  df-sbc 2963  df-csb 3058
This theorem is referenced by: (None)
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