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Mirrors > Home > ILE Home > Th. List > pm13.18 | GIF version |
Description: Theorem *13.18 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.) |
Ref | Expression |
---|---|
pm13.18 | ⊢ ((𝐴 = 𝐵 ∧ 𝐴 ≠ 𝐶) → 𝐵 ≠ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq1 2087 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝐴 = 𝐶 ↔ 𝐵 = 𝐶)) | |
2 | 1 | biimprd 156 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐵 = 𝐶 → 𝐴 = 𝐶)) |
3 | 2 | necon3d 2289 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 ≠ 𝐶 → 𝐵 ≠ 𝐶)) |
4 | 3 | imp 122 | 1 ⊢ ((𝐴 = 𝐵 ∧ 𝐴 ≠ 𝐶) → 𝐵 ≠ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 = wceq 1284 ≠ wne 2245 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-5 1376 ax-gen 1378 ax-4 1440 ax-17 1459 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-cleq 2074 df-ne 2246 |
This theorem is referenced by: pm13.181 2327 |
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