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Theorem sbcne12g 2893
 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 2891 . . 3 (𝐴𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶))
21notbid 600 . 2 (𝐴𝑉 → (¬ [𝐴 / 𝑥]𝐵 = 𝐶 ↔ ¬ 𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶))
3 df-ne 2219 . . . . 5 (𝐵𝐶 ↔ ¬ 𝐵 = 𝐶)
43sbcbii 2842 . . . 4 ([𝐴 / 𝑥]𝐵𝐶[𝐴 / 𝑥] ¬ 𝐵 = 𝐶)
5 sbcng 2823 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥] ¬ 𝐵 = 𝐶 ↔ ¬ [𝐴 / 𝑥]𝐵 = 𝐶))
64, 5syl5bb 185 . . 3 (𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝐶 ↔ ¬ [𝐴 / 𝑥]𝐵 = 𝐶))
7 df-ne 2219 . . . 4 (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶 ↔ ¬ 𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶)
87a1i 9 . . 3 (𝐴𝑉 → (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶 ↔ ¬ 𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶))
96, 8bibi12d 228 . 2 (𝐴𝑉 → (([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶) ↔ (¬ [𝐴 / 𝑥]𝐵 = 𝐶 ↔ ¬ 𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶)))
102, 9mpbird 160 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 102   = wceq 1257   ∈ wcel 1407   ≠ wne 2218  [wsbc 2784  ⦋csb 2877 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 552  ax-in2 553  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036 This theorem depends on definitions:  df-bi 114  df-tru 1260  df-fal 1263  df-nf 1364  df-sb 1660  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-ne 2219  df-v 2574  df-sbc 2785  df-csb 2878 This theorem is referenced by: (None)
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