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| Mirrors > Home > MPE Home > Th. List > bitrd | Structured version Visualization version GIF version | ||
| Description: Deduction form of bitri 275. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Wolf Lammen, 14-Apr-2013.) |
| Ref | Expression |
|---|---|
| bitrd.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
| bitrd.2 | ⊢ (𝜑 → (𝜒 ↔ 𝜃)) |
| Ref | Expression |
|---|---|
| bitrd | ⊢ (𝜑 → (𝜓 ↔ 𝜃)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | bitrd.1 | . . . 4 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
| 2 | 1 | pm5.74i 271 | . . 3 ⊢ ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒)) |
| 3 | bitrd.2 | . . . 4 ⊢ (𝜑 → (𝜒 ↔ 𝜃)) | |
| 4 | 3 | pm5.74i 271 | . . 3 ⊢ ((𝜑 → 𝜒) ↔ (𝜑 → 𝜃)) |
| 5 | 2, 4 | bitri 275 | . 2 ⊢ ((𝜑 → 𝜓) ↔ (𝜑 → 𝜃)) |
| 6 | 5 | pm5.74ri 272 | 1 ⊢ (𝜑 → (𝜓 ↔ 𝜃)) |
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