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| Mirrors > Home > MPE Home > Th. List > pm13.181 | Structured version Visualization version GIF version | ||
| Description: Theorem *13.181 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.) (Proof shortened by Wolf Lammen, 30-Oct-2024.) | 
| Ref | Expression | 
|---|---|
| pm13.181 | ⊢ ((𝐴 = 𝐵 ∧ 𝐵 ≠ 𝐶) → 𝐴 ≠ 𝐶) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | neeq1 3003 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐶)) | |
| 2 | 1 | biimpar 477 | 1 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 ≠ 𝐶) → 𝐴 ≠ 𝐶) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ≠ wne 2940 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-cleq 2729 df-ne 2941 | 
| This theorem is referenced by: fzprval 13625 frgrwopreglem5a 30330 ax6e2ndeqALT 44951 | 
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