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| Mirrors > Home > MPE Home > Th. List > syldan | Structured version Visualization version GIF version | ||
| Description: A syllogism deduction with conjoined antecedents. (Contributed by NM, 24-Feb-2005.) (Proof shortened by Wolf Lammen, 6-Apr-2013.) |
| Ref | Expression |
|---|---|
| syldan.1 | ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
| syldan.2 | ⊢ ((𝜑 ∧ 𝜒) → 𝜃) |
| Ref | Expression |
|---|---|
| syldan | ⊢ ((𝜑 ∧ 𝜓) → 𝜃) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simpl 482 | . 2 ⊢ ((𝜑 ∧ 𝜓) → 𝜑) | |
| 2 | syldan.1 | . 2 ⊢ ((𝜑 ∧ 𝜓) → 𝜒) | |
| 3 | syldan.2 | . 2 ⊢ ((𝜑 ∧ 𝜒) → 𝜃) | |
| 4 | 1, 2, 3 | syl2anc 584 | 1 ⊢ ((𝜑 ∧ 𝜓) → 𝜃) |
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