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| Mirrors > Home > MPE Home > Th. List > sylanbrc | Structured version Visualization version GIF version | ||
| Description: Syllogism inference. (Contributed by Jeff Madsen, 2-Sep-2009.) |
| Ref | Expression |
|---|---|
| sylanbrc.1 | ⊢ (𝜑 → 𝜓) |
| sylanbrc.2 | ⊢ (𝜑 → 𝜒) |
| sylanbrc.3 | ⊢ (𝜃 ↔ (𝜓 ∧ 𝜒)) |
| Ref | Expression |
|---|---|
| sylanbrc | ⊢ (𝜑 → 𝜃) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sylanbrc.1 | . . 3 ⊢ (𝜑 → 𝜓) | |
| 2 | sylanbrc.2 | . . 3 ⊢ (𝜑 → 𝜒) | |
| 3 | 1, 2 | jca 511 | . 2 ⊢ (𝜑 → (𝜓 ∧ 𝜒)) |
| 4 | sylanbrc.3 | . 2 ⊢ (𝜃 ↔ (𝜓 ∧ 𝜒)) | |
| 5 | 3, 4 | sylibr 234 | 1 ⊢ (𝜑 → 𝜃) |
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