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| Mirrors > Home > MPE Home > Th. List > syl2an2r | Structured version Visualization version GIF version | ||
| Description: syl2anr 597 with antecedents in standard conjunction form. (Contributed by Alan Sare, 27-Aug-2016.) (Proof shortened by Wolf Lammen, 28-Mar-2022.) |
| Ref | Expression |
|---|---|
| syl2an2r.1 | ⊢ (𝜑 → 𝜓) |
| syl2an2r.2 | ⊢ ((𝜑 ∧ 𝜒) → 𝜃) |
| syl2an2r.3 | ⊢ ((𝜓 ∧ 𝜃) → 𝜏) |
| Ref | Expression |
|---|---|
| syl2an2r | ⊢ ((𝜑 ∧ 𝜒) → 𝜏) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | syl2an2r.2 | . 2 ⊢ ((𝜑 ∧ 𝜒) → 𝜃) | |
| 2 | syl2an2r.1 | . . 3 ⊢ (𝜑 → 𝜓) | |
| 3 | syl2an2r.3 | . . 3 ⊢ ((𝜓 ∧ 𝜃) → 𝜏) | |
| 4 | 2, 3 | sylan 580 | . 2 ⊢ ((𝜑 ∧ 𝜃) → 𝜏) |
| 5 | 1, 4 | syldan 591 | 1 ⊢ ((𝜑 ∧ 𝜒) → 𝜏) |
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