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Theorem sbcnel12g 3153
 Description: Distribute proper substitution through negated membership. (Contributed by Andrew Salmon, 18-Jun-2011.)
Assertion
Ref Expression
sbcnel12g (A V → ([̣A / xB C[A / x]B [A / x]C))

Proof of Theorem sbcnel12g
StepHypRef Expression
1 df-nel 2519 . . . 4 (B C ↔ ¬ B C)
21sbcbii 3101 . . 3 ([̣A / xB C ↔ [̣A / x]̣ ¬ B C)
32a1i 10 . 2 (A V → ([̣A / xB C ↔ [̣A / x]̣ ¬ B C))
4 sbcng 3086 . 2 (A V → ([̣A / x]̣ ¬ B C ↔ ¬ [̣A / xB C))
5 sbcel12g 3151 . . . 4 (A V → ([̣A / xB C[A / x]B [A / x]C))
65notbid 285 . . 3 (A V → (¬ [̣A / xB C ↔ ¬ [A / x]B [A / x]C))
7 df-nel 2519 . . 3 ([A / x]B [A / x]C ↔ ¬ [A / x]B [A / x]C)
86, 7syl6bbr 254 . 2 (A V → (¬ [̣A / xB C[A / x]B [A / x]C))
93, 4, 83bitrd 270 1 (A V → ([̣A / xB C[A / x]B [A / x]C))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∈ wcel 1710   ∉ wnel 2517  [̣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-nel 2519  df-v 2861  df-sbc 3047  df-csb 3137 This theorem is referenced by: (None)
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