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| Mirrors > Home > MPE Home > Th. List > sylancl | Structured version Visualization version GIF version | ||
| Description: Syllogism inference combined with modus ponens. (Contributed by Jeff Madsen, 2-Sep-2009.) |
| Ref | Expression |
|---|---|
| sylancl.1 | ⊢ (𝜑 → 𝜓) |
| sylancl.2 | ⊢ 𝜒 |
| sylancl.3 | ⊢ ((𝜓 ∧ 𝜒) → 𝜃) |
| Ref | Expression |
|---|---|
| sylancl | ⊢ (𝜑 → 𝜃) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sylancl.1 | . 2 ⊢ (𝜑 → 𝜓) | |
| 2 | sylancl.2 | . . 3 ⊢ 𝜒 | |
| 3 | 2 | a1i 11 | . 2 ⊢ (𝜑 → 𝜒) |
| 4 | sylancl.3 | . 2 ⊢ ((𝜓 ∧ 𝜒) → 𝜃) | |
| 5 | 1, 3, 4 | syl2anc 584 | 1 ⊢ (𝜑 → 𝜃) |
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