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Mirrors > Home > MPE Home > Th. List > sbcne12 | Structured version Visualization version GIF version |
Description: Distribute proper substitution through an inequality. (Contributed by Andrew Salmon, 18-Jun-2011.) (Revised by NM, 18-Aug-2018.) |
Ref | Expression |
---|---|
sbcne12 | ⊢ ([𝐴 / 𝑥]𝐵 ≠ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 ≠ ⦋𝐴 / 𝑥⦌𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nne 2944 | . . . . . 6 ⊢ (¬ 𝐵 ≠ 𝐶 ↔ 𝐵 = 𝐶) | |
2 | 1 | sbcbii 3800 | . . . . 5 ⊢ ([𝐴 / 𝑥] ¬ 𝐵 ≠ 𝐶 ↔ [𝐴 / 𝑥]𝐵 = 𝐶) |
3 | 2 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥] ¬ 𝐵 ≠ 𝐶 ↔ [𝐴 / 𝑥]𝐵 = 𝐶)) |
4 | sbcng 3790 | . . . 4 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥] ¬ 𝐵 ≠ 𝐶 ↔ ¬ [𝐴 / 𝑥]𝐵 ≠ 𝐶)) | |
5 | sbceqg 4370 | . . . . 5 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶)) | |
6 | nne 2944 | . . . . 5 ⊢ (¬ ⦋𝐴 / 𝑥⦌𝐵 ≠ ⦋𝐴 / 𝑥⦌𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶) | |
7 | 5, 6 | bitr4di 289 | . . . 4 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ ¬ ⦋𝐴 / 𝑥⦌𝐵 ≠ ⦋𝐴 / 𝑥⦌𝐶)) |
8 | 3, 4, 7 | 3bitr3d 309 | . . 3 ⊢ (𝐴 ∈ V → (¬ [𝐴 / 𝑥]𝐵 ≠ 𝐶 ↔ ¬ ⦋𝐴 / 𝑥⦌𝐵 ≠ ⦋𝐴 / 𝑥⦌𝐶)) |
9 | 8 | con4bid 317 | . 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝐵 ≠ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 ≠ ⦋𝐴 / 𝑥⦌𝐶)) |
10 | sbcex 3750 | . . . 4 ⊢ ([𝐴 / 𝑥]𝐵 ≠ 𝐶 → 𝐴 ∈ V) | |
11 | 10 | con3i 154 | . . 3 ⊢ (¬ 𝐴 ∈ V → ¬ [𝐴 / 𝑥]𝐵 ≠ 𝐶) |
12 | csbprc 4367 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ∅) | |
13 | csbprc 4367 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐶 = ∅) | |
14 | 12, 13 | eqtr4d 2776 | . . . 4 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶) |
15 | 14, 6 | sylibr 233 | . . 3 ⊢ (¬ 𝐴 ∈ V → ¬ ⦋𝐴 / 𝑥⦌𝐵 ≠ ⦋𝐴 / 𝑥⦌𝐶) |
16 | 11, 15 | 2falsed 377 | . 2 ⊢ (¬ 𝐴 ∈ V → ([𝐴 / 𝑥]𝐵 ≠ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 ≠ ⦋𝐴 / 𝑥⦌𝐶)) |
17 | 9, 16 | pm2.61i 182 | 1 ⊢ ([𝐴 / 𝑥]𝐵 ≠ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 ≠ ⦋𝐴 / 𝑥⦌𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 = wceq 1542 ∈ wcel 2107 ≠ wne 2940 Vcvv 3444 [wsbc 3740 ⦋csb 3856 ∅c0 4283 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-nul 4284 |
This theorem is referenced by: 2nreu 4402 disjdsct 31663 cdlemkid3N 39442 cdlemkid4 39443 |
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