| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > syl2anc | Structured version Visualization version GIF version | ||
| Description: Syllogism inference combined with contraction. (Contributed by NM, 16-Mar-2012.) |
| Ref | Expression |
|---|---|
| syl2anc.1 | ⊢ (𝜑 → 𝜓) |
| syl2anc.2 | ⊢ (𝜑 → 𝜒) |
| syl2anc.3 | ⊢ ((𝜓 ∧ 𝜒) → 𝜃) |
| Ref | Expression |
|---|---|
| syl2anc | ⊢ (𝜑 → 𝜃) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | syl2anc.1 | . 2 ⊢ (𝜑 → 𝜓) | |
| 2 | syl2anc.2 | . 2 ⊢ (𝜑 → 𝜒) | |
| 3 | syl2anc.3 | . . 3 ⊢ ((𝜓 ∧ 𝜒) → 𝜃) | |
| 4 | 3 | ex 412 | . 2 ⊢ (𝜓 → (𝜒 → 𝜃)) |
| 5 | 1, 2, 4 | sylc 65 | 1 ⊢ (𝜑 → 𝜃) |
| Copyright terms: Public domain | W3C validator |