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Mirrors > Home > NFE 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 2359 | . . . 4 ⊢ (A = B → (A = C ↔ B = C)) | |
2 | 1 | biimprd 214 | . . 3 ⊢ (A = B → (B = C → A = C)) |
3 | 2 | necon3d 2554 | . 2 ⊢ (A = B → (A ≠ C → B ≠ C)) |
4 | 3 | imp 418 | 1 ⊢ ((A = B ∧ A ≠ C) → B ≠ C) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 358 = wceq 1642 ≠ wne 2516 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-11 1746 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-an 360 df-ex 1542 df-cleq 2346 df-ne 2518 |
This theorem is referenced by: pm13.181 2589 |
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