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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 2085 | . 2 ⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴) | |
2 | pm13.18 2330 | . 2 ⊢ ((𝐵 = 𝐴 ∧ 𝐵 ≠ 𝐶) → 𝐴 ≠ 𝐶) | |
3 | 1, 2 | sylanb 278 | 1 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 ≠ 𝐶) → 𝐴 ≠ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 = wceq 1285 ≠ wne 2249 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-5 1377 ax-gen 1379 ax-4 1441 ax-17 1460 ax-ext 2065 |
This theorem depends on definitions: df-bi 115 df-cleq 2076 df-ne 2250 |
This theorem is referenced by: fzprval 9211 mod2eq1n2dvds 10470 |
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