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Theorem sbcne12g 2862
 Description: Distribute proper substitution through an inequality. (Contributed by Andrew Salmon, 18-Jun-2011.)
Assertion
Ref Expression
sbcne12g (A 𝑉 → ([A / x]B𝐶A / xBA / x𝐶))

Proof of Theorem sbcne12g
StepHypRef Expression
1 sbceqg 2860 . . 3 (A 𝑉 → ([A / x]B = 𝐶A / xB = A / x𝐶))
21notbid 591 . 2 (A 𝑉 → (¬ [A / x]B = 𝐶 ↔ ¬ A / xB = A / x𝐶))
3 df-ne 2203 . . . . 5 (B𝐶 ↔ ¬ B = 𝐶)
43sbcbii 2812 . . . 4 ([A / x]B𝐶[A / x] ¬ B = 𝐶)
5 sbcng 2797 . . . 4 (A 𝑉 → ([A / x] ¬ B = 𝐶 ↔ ¬ [A / x]B = 𝐶))
64, 5syl5bb 181 . . 3 (A 𝑉 → ([A / x]B𝐶 ↔ ¬ [A / x]B = 𝐶))
7 df-ne 2203 . . . 4 (A / xBA / x𝐶 ↔ ¬ A / xB = A / x𝐶)
87a1i 9 . . 3 (A 𝑉 → (A / xBA / x𝐶 ↔ ¬ A / xB = A / x𝐶))
96, 8bibi12d 224 . 2 (A 𝑉 → (([A / x]B𝐶A / xBA / x𝐶) ↔ (¬ [A / x]B = 𝐶 ↔ ¬ A / xB = A / x𝐶)))
102, 9mpbird 156 1 (A 𝑉 → ([A / x]B𝐶A / xBA / x𝐶))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 98   = wceq 1242   ∈ wcel 1390   ≠ wne 2201  [wsbc 2758  ⦋csb 2846 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-v 2553  df-sbc 2759  df-csb 2847 This theorem is referenced by: (None)
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