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Theorem sbcne12g 3154
 Description: Distribute proper substitution through an inequality. (Contributed by Andrew Salmon, 18-Jun-2011.)
Assertion
Ref Expression
sbcne12g (A V → ([̣A / xBC[A / x]B[A / x]C))

Proof of Theorem sbcne12g
StepHypRef Expression
1 nne 2520 . . . . 5 BCB = C)
21sbcbii 3101 . . . 4 ([̣A / x]̣ ¬ BC ↔ [̣A / xB = C)
32a1i 10 . . 3 (A V → ([̣A / x]̣ ¬ BC ↔ [̣A / xB = C))
4 sbcng 3086 . . 3 (A V → ([̣A / x]̣ ¬ BC ↔ ¬ [̣A / xBC))
5 sbceqg 3152 . . . 4 (A V → ([̣A / xB = C[A / x]B = [A / x]C))
6 nne 2520 . . . 4 [A / x]B[A / x]C[A / x]B = [A / x]C)
75, 6syl6bbr 254 . . 3 (A V → ([̣A / xB = C ↔ ¬ [A / x]B[A / x]C))
83, 4, 73bitr3d 274 . 2 (A V → (¬ [̣A / xBC ↔ ¬ [A / x]B[A / x]C))
98con4bid 284 1 (A V → ([̣A / xBC[A / x]B[A / x]C))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   = wceq 1642   ∈ wcel 1710   ≠ wne 2516  [̣wsbc 3046  [csb 3136 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-v 2861  df-sbc 3047  df-csb 3137 This theorem is referenced by: (None)
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