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Theorem sbcne12 3937
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 2785 . . . . . 6 𝐵𝐶𝐵 = 𝐶)
21sbcbii 3457 . . . . 5 ([𝐴 / 𝑥] ¬ 𝐵𝐶[𝐴 / 𝑥]𝐵 = 𝐶)
32a1i 11 . . . 4 (𝐴 ∈ V → ([𝐴 / 𝑥] ¬ 𝐵𝐶[𝐴 / 𝑥]𝐵 = 𝐶))
4 sbcng 3442 . . . 4 (𝐴 ∈ V → ([𝐴 / 𝑥] ¬ 𝐵𝐶 ↔ ¬ [𝐴 / 𝑥]𝐵𝐶))
5 sbceqg 3935 . . . . 5 (𝐴 ∈ V → ([𝐴 / 𝑥]𝐵 = 𝐶𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶))
6 nne 2785 . . . . 5 𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶)
75, 6syl6bbr 276 . . . 4 (𝐴 ∈ V → ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ ¬ 𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
83, 4, 73bitr3d 296 . . 3 (𝐴 ∈ V → (¬ [𝐴 / 𝑥]𝐵𝐶 ↔ ¬ 𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
98con4bid 305 . 2 (𝐴 ∈ V → ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
10 sbcex 3411 . . . 4 ([𝐴 / 𝑥]𝐵𝐶𝐴 ∈ V)
1110con3i 148 . . 3 𝐴 ∈ V → ¬ [𝐴 / 𝑥]𝐵𝐶)
12 csbprc 3931 . . . . 5 𝐴 ∈ V → 𝐴 / 𝑥𝐵 = ∅)
13 csbprc 3931 . . . . 5 𝐴 ∈ V → 𝐴 / 𝑥𝐶 = ∅)
1412, 13eqtr4d 2646 . . . 4 𝐴 ∈ V → 𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐶)
1514, 6sylibr 222 . . 3 𝐴 ∈ V → ¬ 𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)
1611, 152falsed 364 . 2 𝐴 ∈ V → ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
179, 16pm2.61i 174 1 ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 194   = wceq 1474  wcel 1976  wne 2779  Vcvv 3172  [wsbc 3401  csb 3498  c0 3873
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-nul 3874
This theorem is referenced by:  disjdsct  28697  cdlemkid3N  35063  cdlemkid4  35064
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