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Theorem sbcne12 4310
 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 2956 . . . . . 6 𝐵𝐶𝐵 = 𝐶)
21sbcbii 3754 . . . . 5 ([𝐴 / 𝑥] ¬ 𝐵𝐶[𝐴 / 𝑥]𝐵 = 𝐶)
32a1i 11 . . . 4 (𝐴 ∈ V → ([𝐴 / 𝑥] ¬ 𝐵𝐶[𝐴 / 𝑥]𝐵 = 𝐶))
4 sbcng 3744 . . . 4 (𝐴 ∈ V → ([𝐴 / 𝑥] ¬ 𝐵𝐶 ↔ ¬ [𝐴 / 𝑥]𝐵𝐶))
5 sbceqg 4307 . . . . 5 (𝐴 ∈ V → ([𝐴 / 𝑥]𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶))
6 nne 2956 . . . . 5 𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶)
75, 6bitr4di 293 . . . 4 (𝐴 ∈ V → ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ ¬ 𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
83, 4, 73bitr3d 313 . . 3 (𝐴 ∈ V → (¬ [𝐴 / 𝑥]𝐵𝐶 ↔ ¬ 𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
98con4bid 321 . 2 (𝐴 ∈ V → ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
10 sbcex 3707 . . . 4 ([𝐴 / 𝑥]𝐵𝐶𝐴 ∈ V)
1110con3i 157 . . 3 𝐴 ∈ V → ¬ [𝐴 / 𝑥]𝐵𝐶)
12 csbprc 4303 . . . . 5 𝐴 ∈ V → 𝐴 / 𝑥𝐵 = ∅)
13 csbprc 4303 . . . . 5 𝐴 ∈ V → 𝐴 / 𝑥𝐶 = ∅)
1412, 13eqtr4d 2797 . . . 4 𝐴 ∈ V → 𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶)
1514, 6sylibr 237 . . 3 𝐴 ∈ V → ¬ 𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)
1611, 152falsed 381 . 2 𝐴 ∈ V → ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
179, 16pm2.61i 185 1 ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 209   = wceq 1539   ∈ wcel 2112   ≠ wne 2952  Vcvv 3410  [wsbc 3697  ⦋csb 3806  ∅c0 4226 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-nul 4227 This theorem is referenced by:  2nreu  4339  disjdsct  30560  cdlemkid3N  38510  cdlemkid4  38511
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