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Theorem sbcne12g 2868
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 2866 . . 3 (𝐴𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶))
21notbid 592 . 2 (𝐴𝑉 → (¬ [𝐴 / 𝑥]𝐵 = 𝐶 ↔ ¬ 𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶))
3 df-ne 2206 . . . . 5 (𝐵𝐶 ↔ ¬ 𝐵 = 𝐶)
43sbcbii 2818 . . . 4 ([𝐴 / 𝑥]𝐵𝐶[𝐴 / 𝑥] ¬ 𝐵 = 𝐶)
5 sbcng 2803 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥] ¬ 𝐵 = 𝐶 ↔ ¬ [𝐴 / 𝑥]𝐵 = 𝐶))
64, 5syl5bb 181 . . 3 (𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝐶 ↔ ¬ [𝐴 / 𝑥]𝐵 = 𝐶))
7 df-ne 2206 . . . 4 (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶 ↔ ¬ 𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶)
87a1i 9 . . 3 (𝐴𝑉 → (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶 ↔ ¬ 𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶))
96, 8bibi12d 224 . 2 (𝐴𝑉 → (([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶) ↔ (¬ [𝐴 / 𝑥]𝐵 = 𝐶 ↔ ¬ 𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶)))
102, 9mpbird 156 1 (𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 98   = wceq 1243  wcel 1393  wne 2204  [wsbc 2764  csb 2852
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-v 2559  df-sbc 2765  df-csb 2853
This theorem is referenced by: (None)
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