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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 3826 | . 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥] ¬ 𝐵 ∈ 𝐶 ↔ ¬ [𝐴 / 𝑥]𝐵 ∈ 𝐶)) | |
2 | df-nel 3037 | . . 3 ⊢ (𝐵 ∉ 𝐶 ↔ ¬ 𝐵 ∈ 𝐶) | |
3 | 2 | sbcbii 3836 | . 2 ⊢ ([𝐴 / 𝑥]𝐵 ∉ 𝐶 ↔ [𝐴 / 𝑥] ¬ 𝐵 ∈ 𝐶) |
4 | df-nel 3037 | . . 3 ⊢ (⦋𝐴 / 𝑥⦌𝐵 ∉ ⦋𝐴 / 𝑥⦌𝐶 ↔ ¬ ⦋𝐴 / 𝑥⦌𝐵 ∈ ⦋𝐴 / 𝑥⦌𝐶) | |
5 | sbcel12 4405 | . . 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 2099 ∉ wnel 3036 [wsbc 3775 ⦋csb 3891 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-nel 3037 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-nul 4323 |
This theorem is referenced by: rusbcALT 44150 |
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