Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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, 14-May-2023.) |
Ref | Expression |
---|---|
pm13.18 | ⊢ ((𝐴 = 𝐵 ∧ 𝐴 ≠ 𝐶) → 𝐵 ≠ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neeq1 3078 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐶)) | |
2 | 1 | biimpd 230 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 ≠ 𝐶 → 𝐵 ≠ 𝐶)) |
3 | 2 | imp 407 | 1 ⊢ ((𝐴 = 𝐵 ∧ 𝐴 ≠ 𝐶) → 𝐵 ≠ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ≠ wne 3016 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-9 2115 ax-ext 2793 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1772 df-cleq 2814 df-ne 3017 |
This theorem is referenced by: pm13.181 3099 iotan0 6339 frgrwopreglem5a 28018 4atexlemex4 37091 cncfiooicclem1 42056 |
Copyright terms: Public domain | W3C validator |