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| Mirrors > Home > MPE Home > Th. List > impbid2 | Structured version Visualization version GIF version | ||
| Description: Infer an equivalence from two implications. (Contributed by NM, 6-Mar-2007.) (Proof shortened by Wolf Lammen, 27-Sep-2013.) |
| Ref | Expression |
|---|---|
| impbid2.1 | ⊢ (𝜓 → 𝜒) |
| impbid2.2 | ⊢ (𝜑 → (𝜒 → 𝜓)) |
| Ref | Expression |
|---|---|
| impbid2 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | impbid2.2 | . . 3 ⊢ (𝜑 → (𝜒 → 𝜓)) | |
| 2 | impbid2.1 | . . 3 ⊢ (𝜓 → 𝜒) | |
| 3 | 1, 2 | impbid1 225 | . 2 ⊢ (𝜑 → (𝜒 ↔ 𝜓)) |
| 4 | 3 | bicomd 223 | 1 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
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