![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > sbcne12g | GIF version |
Description: Distribute proper substitution through an inequality. (Contributed by Andrew Salmon, 18-Jun-2011.) |
Ref | Expression |
---|---|
sbcne12g | ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 ≠ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 ≠ ⦋𝐴 / 𝑥⦌𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbceqg 3073 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶)) | |
2 | 1 | notbid 667 | . 2 ⊢ (𝐴 ∈ 𝑉 → (¬ [𝐴 / 𝑥]𝐵 = 𝐶 ↔ ¬ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶)) |
3 | df-ne 2348 | . . . . 5 ⊢ (𝐵 ≠ 𝐶 ↔ ¬ 𝐵 = 𝐶) | |
4 | 3 | sbcbii 3022 | . . . 4 ⊢ ([𝐴 / 𝑥]𝐵 ≠ 𝐶 ↔ [𝐴 / 𝑥] ¬ 𝐵 = 𝐶) |
5 | sbcng 3003 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥] ¬ 𝐵 = 𝐶 ↔ ¬ [𝐴 / 𝑥]𝐵 = 𝐶)) | |
6 | 4, 5 | bitrid 192 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 ≠ 𝐶 ↔ ¬ [𝐴 / 𝑥]𝐵 = 𝐶)) |
7 | df-ne 2348 | . . . 4 ⊢ (⦋𝐴 / 𝑥⦌𝐵 ≠ ⦋𝐴 / 𝑥⦌𝐶 ↔ ¬ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶) | |
8 | 7 | a1i 9 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌𝐵 ≠ ⦋𝐴 / 𝑥⦌𝐶 ↔ ¬ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶)) |
9 | 6, 8 | bibi12d 235 | . 2 ⊢ (𝐴 ∈ 𝑉 → (([𝐴 / 𝑥]𝐵 ≠ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 ≠ ⦋𝐴 / 𝑥⦌𝐶) ↔ (¬ [𝐴 / 𝑥]𝐵 = 𝐶 ↔ ¬ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶))) |
10 | 2, 9 | mpbird 167 | 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 105 = wceq 1353 ∈ wcel 2148 ≠ wne 2347 [wsbc 2962 ⦋csb 3057 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-v 2739 df-sbc 2963 df-csb 3058 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |