| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > impbid1 | Structured version Visualization version GIF version | ||
| Description: Infer an equivalence from two implications. (Contributed by NM, 6-Mar-2007.) |
| Ref | Expression |
|---|---|
| impbid1.1 | ⊢ (𝜑 → (𝜓 → 𝜒)) |
| impbid1.2 | ⊢ (𝜒 → 𝜓) |
| Ref | Expression |
|---|---|
| impbid1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | impbid1.1 | . 2 ⊢ (𝜑 → (𝜓 → 𝜒)) | |
| 2 | impbid1.2 | . . 3 ⊢ (𝜒 → 𝜓) | |
| 3 | 2 | a1i 11 | . 2 ⊢ (𝜑 → (𝜒 → 𝜓)) |
| 4 | 1, 3 | impbid 212 | 1 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
| Copyright terms: Public domain | W3C validator |