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| Mirrors > Home > MPE Home > Th. List > syl3anbrc | Structured version Visualization version GIF version | ||
| Description: Syllogism inference. (Contributed by Mario Carneiro, 11-May-2014.) |
| Ref | Expression |
|---|---|
| syl3anbrc.1 | ⊢ (𝜑 → 𝜓) |
| syl3anbrc.2 | ⊢ (𝜑 → 𝜒) |
| syl3anbrc.3 | ⊢ (𝜑 → 𝜃) |
| syl3anbrc.4 | ⊢ (𝜏 ↔ (𝜓 ∧ 𝜒 ∧ 𝜃)) |
| Ref | Expression |
|---|---|
| syl3anbrc | ⊢ (𝜑 → 𝜏) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | syl3anbrc.1 | . . 3 ⊢ (𝜑 → 𝜓) | |
| 2 | syl3anbrc.2 | . . 3 ⊢ (𝜑 → 𝜒) | |
| 3 | syl3anbrc.3 | . . 3 ⊢ (𝜑 → 𝜃) | |
| 4 | 1, 2, 3 | 3jca 1129 | . 2 ⊢ (𝜑 → (𝜓 ∧ 𝜒 ∧ 𝜃)) |
| 5 | syl3anbrc.4 | . 2 ⊢ (𝜏 ↔ (𝜓 ∧ 𝜒 ∧ 𝜃)) | |
| 6 | 4, 5 | sylibr 234 | 1 ⊢ (𝜑 → 𝜏) |
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