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| Mirrors > Home > MPE Home > Th. List > sylan | Structured version Visualization version GIF version | ||
| Description: A syllogism inference. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 22-Nov-2012.) |
| Ref | Expression |
|---|---|
| sylan.1 | ⊢ (𝜑 → 𝜓) |
| sylan.2 | ⊢ ((𝜓 ∧ 𝜒) → 𝜃) |
| Ref | Expression |
|---|---|
| sylan | ⊢ ((𝜑 ∧ 𝜒) → 𝜃) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sylan.1 | . 2 ⊢ (𝜑 → 𝜓) | |
| 2 | sylan.2 | . . 3 ⊢ ((𝜓 ∧ 𝜒) → 𝜃) | |
| 3 | 2 | expcom 413 | . 2 ⊢ (𝜒 → (𝜓 → 𝜃)) |
| 4 | 1, 3 | mpan9 506 | 1 ⊢ ((𝜑 ∧ 𝜒) → 𝜃) |
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