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Mirrors > Home > MPE Home > Th. List > sbcnel12g | Structured version Visualization version 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 | sbcng 3789 | . 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥] ¬ 𝐵 ∈ 𝐶 ↔ ¬ [𝐴 / 𝑥]𝐵 ∈ 𝐶)) | |
2 | df-nel 3050 | . . 3 ⊢ (𝐵 ∉ 𝐶 ↔ ¬ 𝐵 ∈ 𝐶) | |
3 | 2 | sbcbii 3799 | . 2 ⊢ ([𝐴 / 𝑥]𝐵 ∉ 𝐶 ↔ [𝐴 / 𝑥] ¬ 𝐵 ∈ 𝐶) |
4 | df-nel 3050 | . . 3 ⊢ (⦋𝐴 / 𝑥⦌𝐵 ∉ ⦋𝐴 / 𝑥⦌𝐶 ↔ ¬ ⦋𝐴 / 𝑥⦌𝐵 ∈ ⦋𝐴 / 𝑥⦌𝐶) | |
5 | sbcel12 4368 | . . 3 ⊢ ([𝐴 / 𝑥]𝐵 ∈ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 ∈ ⦋𝐴 / 𝑥⦌𝐶) | |
6 | 4, 5 | xchbinxr 334 | . 2 ⊢ (⦋𝐴 / 𝑥⦌𝐵 ∉ ⦋𝐴 / 𝑥⦌𝐶 ↔ ¬ [𝐴 / 𝑥]𝐵 ∈ 𝐶) |
7 | 1, 3, 6 | 3bitr4g 313 | 1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 ∉ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 ∉ ⦋𝐴 / 𝑥⦌𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∈ wcel 2106 ∉ wnel 3049 [wsbc 3739 ⦋csb 3855 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-nel 3050 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-nul 4283 |
This theorem is referenced by: rusbcALT 42700 |
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