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| Mirrors > Home > MPE Home > Th. List > mpan | Structured version Visualization version GIF version | ||
| Description: An inference based on modus ponens. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Wolf Lammen, 7-Apr-2013.) |
| Ref | Expression |
|---|---|
| mpan.1 | ⊢ 𝜑 |
| mpan.2 | ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
| Ref | Expression |
|---|---|
| mpan | ⊢ (𝜓 → 𝜒) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mpan.1 | . . 3 ⊢ 𝜑 | |
| 2 | 1 | a1i 11 | . 2 ⊢ (𝜓 → 𝜑) |
| 3 | mpan.2 | . 2 ⊢ ((𝜑 ∧ 𝜓) → 𝜒) | |
| 4 | 2, 3 | mpancom 688 | 1 ⊢ (𝜓 → 𝜒) |
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