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Mirrors > Home > ILE Home > Th. List > sbcnel12g | GIF version |
Description: Distribute proper substitution through negated membership. (Contributed by Andrew Salmon, 18-Jun-2011.) |
Ref | Expression |
---|---|
sbcnel12g | ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 ∉ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 ∉ ⦋𝐴 / 𝑥⦌𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nel 2381 | . . . 4 ⊢ (𝐵 ∉ 𝐶 ↔ ¬ 𝐵 ∈ 𝐶) | |
2 | 1 | sbcbii 2940 | . . 3 ⊢ ([𝐴 / 𝑥]𝐵 ∉ 𝐶 ↔ [𝐴 / 𝑥] ¬ 𝐵 ∈ 𝐶) |
3 | 2 | a1i 9 | . 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 ∉ 𝐶 ↔ [𝐴 / 𝑥] ¬ 𝐵 ∈ 𝐶)) |
4 | sbcng 2921 | . 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥] ¬ 𝐵 ∈ 𝐶 ↔ ¬ [𝐴 / 𝑥]𝐵 ∈ 𝐶)) | |
5 | sbcel12g 2988 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 ∈ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 ∈ ⦋𝐴 / 𝑥⦌𝐶)) | |
6 | 5 | notbid 641 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (¬ [𝐴 / 𝑥]𝐵 ∈ 𝐶 ↔ ¬ ⦋𝐴 / 𝑥⦌𝐵 ∈ ⦋𝐴 / 𝑥⦌𝐶)) |
7 | df-nel 2381 | . . 3 ⊢ (⦋𝐴 / 𝑥⦌𝐵 ∉ ⦋𝐴 / 𝑥⦌𝐶 ↔ ¬ ⦋𝐴 / 𝑥⦌𝐵 ∈ ⦋𝐴 / 𝑥⦌𝐶) | |
8 | 6, 7 | syl6bbr 197 | . 2 ⊢ (𝐴 ∈ 𝑉 → (¬ [𝐴 / 𝑥]𝐵 ∈ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 ∉ ⦋𝐴 / 𝑥⦌𝐶)) |
9 | 3, 4, 8 | 3bitrd 213 | 1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 ∉ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 ∉ ⦋𝐴 / 𝑥⦌𝐶)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 104 ∈ wcel 1465 ∉ wnel 2380 [wsbc 2882 ⦋csb 2975 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-nel 2381 df-v 2662 df-sbc 2883 df-csb 2976 |
This theorem is referenced by: (None) |
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