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| Mirrors > Home > MPE Home > Th. List > mpdan | Structured version Visualization version GIF version | ||
| Description: An inference based on modus ponens. (Contributed by NM, 23-May-1999.) (Proof shortened by Wolf Lammen, 22-Nov-2012.) |
| Ref | Expression |
|---|---|
| mpdan.1 | ⊢ (𝜑 → 𝜓) |
| mpdan.2 | ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
| Ref | Expression |
|---|---|
| mpdan | ⊢ (𝜑 → 𝜒) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | id 22 | . 2 ⊢ (𝜑 → 𝜑) | |
| 2 | mpdan.1 | . 2 ⊢ (𝜑 → 𝜓) | |
| 3 | mpdan.2 | . 2 ⊢ ((𝜑 ∧ 𝜓) → 𝜒) | |
| 4 | 1, 2, 3 | syl2anc 584 | 1 ⊢ (𝜑 → 𝜒) |
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