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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.) |
Ref | Expression |
---|---|
pm13.181 | ⊢ ((𝐴 = 𝐵 ∧ 𝐵 ≠ 𝐶) → 𝐴 ≠ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqcom 2745 | . 2 ⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴) | |
2 | pm13.18 3015 | . 2 ⊢ ((𝐵 = 𝐴 ∧ 𝐵 ≠ 𝐶) → 𝐴 ≠ 𝐶) | |
3 | 1, 2 | sylanb 584 | 1 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 ≠ 𝐶) → 𝐴 ≠ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ≠ wne 2934 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-9 2124 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1787 df-cleq 2730 df-ne 2935 |
This theorem is referenced by: fzprval 13059 frgrwopreglem5a 28248 ax6e2ndeqALT 42089 |
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