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Theorem sbcnel12g 2861
Description: Distribute proper substitution through negated membership. (Contributed by Andrew Salmon, 18-Jun-2011.)
Assertion
Ref Expression
sbcnel12g (A 𝑉 → ([A / x]B𝐶A / xBA / x𝐶))

Proof of Theorem sbcnel12g
StepHypRef Expression
1 df-nel 2204 . . . 4 (B𝐶 ↔ ¬ B 𝐶)
21sbcbii 2812 . . 3 ([A / x]B𝐶[A / x] ¬ B 𝐶)
32a1i 9 . 2 (A 𝑉 → ([A / x]B𝐶[A / x] ¬ B 𝐶))
4 sbcng 2797 . 2 (A 𝑉 → ([A / x] ¬ B 𝐶 ↔ ¬ [A / x]B 𝐶))
5 sbcel12g 2859 . . . 4 (A 𝑉 → ([A / x]B 𝐶A / xB A / x𝐶))
65notbid 591 . . 3 (A 𝑉 → (¬ [A / x]B 𝐶 ↔ ¬ A / xB A / x𝐶))
7 df-nel 2204 . . 3 (A / xBA / x𝐶 ↔ ¬ A / xB A / x𝐶)
86, 7syl6bbr 187 . 2 (A 𝑉 → (¬ [A / x]B 𝐶A / xBA / x𝐶))
93, 4, 83bitrd 203 1 (A 𝑉 → ([A / x]B𝐶A / xBA / x𝐶))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 98   wcel 1390  wnel 2202  [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-nel 2204  df-v 2553  df-sbc 2759  df-csb 2847
This theorem is referenced by: (None)
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