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Mirrors > Home > MPE Home > Th. List > impbid21d | Structured version Visualization version GIF version |
Description: Deduce an equivalence from two implications. (Contributed by Wolf Lammen, 12-May-2013.) |
Ref | Expression |
---|---|
impbid21d.1 | ⊢ (𝜓 → (𝜒 → 𝜃)) |
impbid21d.2 | ⊢ (𝜑 → (𝜃 → 𝜒)) |
Ref | Expression |
---|---|
impbid21d | ⊢ (𝜑 → (𝜓 → (𝜒 ↔ 𝜃))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | impbid21d.1 | . . 3 ⊢ (𝜓 → (𝜒 → 𝜃)) | |
2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃))) |
3 | impbid21d.2 | . . 3 ⊢ (𝜑 → (𝜃 → 𝜒)) | |
4 | 3 | a1d 25 | . 2 ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜒))) |
5 | 2, 4 | impbidd 199 | 1 ⊢ (𝜑 → (𝜓 → (𝜒 ↔ 𝜃))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 196 |
This theorem is referenced by: impbid 201 pm5.1im 252 rp-fakenanass 36879 |
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