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| Mirrors > Home > MPE Home > Th. List > impbii | Structured version Visualization version GIF version | ||
| Description: Infer an equivalence from an implication and its converse. Inference associated with impbi 208. (Contributed by NM, 29-Dec-1992.) |
| Ref | Expression |
|---|---|
| impbii.1 | ⊢ (𝜑 → 𝜓) |
| impbii.2 | ⊢ (𝜓 → 𝜑) |
| Ref | Expression |
|---|---|
| impbii | ⊢ (𝜑 ↔ 𝜓) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | impbii.1 | . 2 ⊢ (𝜑 → 𝜓) | |
| 2 | impbii.2 | . 2 ⊢ (𝜓 → 𝜑) | |
| 3 | impbi 208 | . 2 ⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜑) → (𝜑 ↔ 𝜓))) | |
| 4 | 1, 2, 3 | mp2 9 | 1 ⊢ (𝜑 ↔ 𝜓) |
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