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| Mirrors > Home > MPE Home > Th. List > bitri | Structured version Visualization version GIF version | ||
| Description: An inference from transitive law for logical equivalence. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 13-Oct-2012.) |
| Ref | Expression |
|---|---|
| bitri.1 | ⊢ (𝜑 ↔ 𝜓) |
| bitri.2 | ⊢ (𝜓 ↔ 𝜒) |
| Ref | Expression |
|---|---|
| bitri | ⊢ (𝜑 ↔ 𝜒) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | bitri.1 | . . 3 ⊢ (𝜑 ↔ 𝜓) | |
| 2 | bitri.2 | . . 3 ⊢ (𝜓 ↔ 𝜒) | |
| 3 | 1, 2 | sylbb 219 | . 2 ⊢ (𝜑 → 𝜒) |
| 4 | 1, 2 | sylbbr 236 | . 2 ⊢ (𝜒 → 𝜑) |
| 5 | 3, 4 | impbii 209 | 1 ⊢ (𝜑 ↔ 𝜒) |
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