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Mirrors > Home > ILE Home > Th. List > pm13.181 | GIF version |
Description: Theorem *13.181 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.) |
Ref | Expression |
---|---|
pm13.181 | ⊢ ((𝐴 = 𝐵 ∧ 𝐵 ≠ 𝐶) → 𝐴 ≠ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqcom 2042 | . 2 ⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴) | |
2 | pm13.18 2286 | . 2 ⊢ ((𝐵 = 𝐴 ∧ 𝐵 ≠ 𝐶) → 𝐴 ≠ 𝐶) | |
3 | 1, 2 | sylanb 268 | 1 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 ≠ 𝐶) → 𝐴 ≠ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1243 ≠ wne 2204 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-5 1336 ax-gen 1338 ax-4 1400 ax-17 1419 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-cleq 2033 df-ne 2206 |
This theorem is referenced by: fzprval 8944 |
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