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Mirrors > Home > MPE Home > Th. List > impbida | Structured version Visualization version GIF version |
Description: Deduce an equivalence from two implications. Variant of impbid 215. (Contributed by NM, 17-Feb-2007.) |
Ref | Expression |
---|---|
impbida.1 | ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
impbida.2 | ⊢ ((𝜑 ∧ 𝜒) → 𝜓) |
Ref | Expression |
---|---|
impbida | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | impbida.1 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜒) | |
2 | 1 | ex 416 | . 2 ⊢ (𝜑 → (𝜓 → 𝜒)) |
3 | impbida.2 | . . 3 ⊢ ((𝜑 ∧ 𝜒) → 𝜓) | |
4 | 3 | ex 416 | . 2 ⊢ (𝜑 → (𝜒 → 𝜓)) |
5 | 2, 4 | impbid 215 | 1 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
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