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Mirrors > Home > MPE Home > Th. List > pm13.18 | Structured version Visualization version 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 2614 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝐴 = 𝐶 ↔ 𝐵 = 𝐶)) | |
2 | 1 | biimprd 237 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐵 = 𝐶 → 𝐴 = 𝐶)) |
3 | 2 | necon3d 2803 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 ≠ 𝐶 → 𝐵 ≠ 𝐶)) |
4 | 3 | imp 444 | 1 ⊢ ((𝐴 = 𝐵 ∧ 𝐴 ≠ 𝐶) → 𝐵 ≠ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ≠ wne 2780 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-an 385 df-cleq 2603 df-ne 2782 |
This theorem is referenced by: pm13.181 2864 4atexlemex4 34377 cncfiooicclem1 38779 |
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