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Theorem sbcne12 4416
Description: Distribute proper substitution through an inequality. (Contributed by Andrew Salmon, 18-Jun-2011.) (Revised by NM, 18-Aug-2018.)
Assertion
Ref Expression
sbcne12 ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)

Proof of Theorem sbcne12
StepHypRef Expression
1 nne 2941 . . . . . 6 𝐵𝐶𝐵 = 𝐶)
21sbcbii 3839 . . . . 5 ([𝐴 / 𝑥] ¬ 𝐵𝐶[𝐴 / 𝑥]𝐵 = 𝐶)
32a1i 11 . . . 4 (𝐴 ∈ V → ([𝐴 / 𝑥] ¬ 𝐵𝐶[𝐴 / 𝑥]𝐵 = 𝐶))
4 sbcng 3829 . . . 4 (𝐴 ∈ V → ([𝐴 / 𝑥] ¬ 𝐵𝐶 ↔ ¬ [𝐴 / 𝑥]𝐵𝐶))
5 sbceqg 4413 . . . . 5 (𝐴 ∈ V → ([𝐴 / 𝑥]𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶))
6 nne 2941 . . . . 5 𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶)
75, 6bitr4di 288 . . . 4 (𝐴 ∈ V → ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ ¬ 𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
83, 4, 73bitr3d 308 . . 3 (𝐴 ∈ V → (¬ [𝐴 / 𝑥]𝐵𝐶 ↔ ¬ 𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
98con4bid 316 . 2 (𝐴 ∈ V → ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
10 sbcex 3788 . . . 4 ([𝐴 / 𝑥]𝐵𝐶𝐴 ∈ V)
1110con3i 154 . . 3 𝐴 ∈ V → ¬ [𝐴 / 𝑥]𝐵𝐶)
12 csbprc 4410 . . . . 5 𝐴 ∈ V → 𝐴 / 𝑥𝐵 = ∅)
13 csbprc 4410 . . . . 5 𝐴 ∈ V → 𝐴 / 𝑥𝐶 = ∅)
1412, 13eqtr4d 2771 . . . 4 𝐴 ∈ V → 𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶)
1514, 6sylibr 233 . . 3 𝐴 ∈ V → ¬ 𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)
1611, 152falsed 375 . 2 𝐴 ∈ V → ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
179, 16pm2.61i 182 1 ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205   = wceq 1533  wcel 2098  wne 2937  Vcvv 3473  [wsbc 3778  csb 3894  c0 4326
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-nul 4327
This theorem is referenced by:  2nreu  4445  disjdsct  32503  cdlemkid3N  40438  cdlemkid4  40439
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