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| Mirrors > Home > MPE Home > Th. List > syl2an | Structured version Visualization version GIF version | ||
| Description: A double syllogism inference. For an implication-only version, see syl2im 40. (Contributed by NM, 31-Jan-1997.) |
| Ref | Expression |
|---|---|
| syl2an.1 | ⊢ (𝜑 → 𝜓) |
| syl2an.2 | ⊢ (𝜏 → 𝜒) |
| syl2an.3 | ⊢ ((𝜓 ∧ 𝜒) → 𝜃) |
| Ref | Expression |
|---|---|
| syl2an | ⊢ ((𝜑 ∧ 𝜏) → 𝜃) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | syl2an.2 | . 2 ⊢ (𝜏 → 𝜒) | |
| 2 | syl2an.1 | . . 3 ⊢ (𝜑 → 𝜓) | |
| 3 | syl2an.3 | . . 3 ⊢ ((𝜓 ∧ 𝜒) → 𝜃) | |
| 4 | 2, 3 | sylan 580 | . 2 ⊢ ((𝜑 ∧ 𝜒) → 𝜃) |
| 5 | 1, 4 | sylan2 593 | 1 ⊢ ((𝜑 ∧ 𝜏) → 𝜃) |
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