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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.) (Proof shortened by Wolf Lammen, 29-Oct-2024.) |
Ref | Expression |
---|---|
pm13.18 | ⊢ ((𝐴 = 𝐵 ∧ 𝐴 ≠ 𝐶) → 𝐵 ≠ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neeq1 2997 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐶)) | |
2 | 1 | biimpa 476 | 1 ⊢ ((𝐴 = 𝐵 ∧ 𝐴 ≠ 𝐶) → 𝐵 ≠ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ≠ wne 2934 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-9 2108 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1774 df-cleq 2718 df-ne 2935 |
This theorem is referenced by: pm13.181OLD 3018 iotan0 6526 frgrwopreglem5a 30069 4atexlemex4 39455 cncfiooicclem1 45162 |
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