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| Mirrors > Home > MPE Home > Th. List > syl3an1 | Structured version Visualization version GIF version | ||
| Description: A syllogism inference. (Contributed by NM, 22-Aug-1995.) |
| Ref | Expression |
|---|---|
| syl3an1.1 | ⊢ (𝜑 → 𝜓) |
| syl3an1.2 | ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏) |
| Ref | Expression |
|---|---|
| syl3an1 | ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃) → 𝜏) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | syl3an1.1 | . . 3 ⊢ (𝜑 → 𝜓) | |
| 2 | 1 | 3anim1i 1153 | . 2 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃) → (𝜓 ∧ 𝜒 ∧ 𝜃)) |
| 3 | syl3an1.2 | . 2 ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → 𝜏) | |
| 4 | 2, 3 | syl 17 | 1 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃) → 𝜏) |
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