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Mirrors > Home > MPE Home > Th. List > imbi12 | Structured version Visualization version GIF version |
Description: Closed form of imbi12i 343. Was automatically derived from its "Virtual Deduction" version and the Metamath program "MM-PA> MINIMIZE_WITH *" command. (Contributed by Alan Sare, 18-Mar-2012.) |
Ref | Expression |
---|---|
imbi12 | ⊢ ((𝜑 ↔ 𝜓) → ((𝜒 ↔ 𝜃) → ((𝜑 → 𝜒) ↔ (𝜓 → 𝜃)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simplim 165 | . . 3 ⊢ (¬ ((𝜑 ↔ 𝜓) → ¬ (𝜒 ↔ 𝜃)) → (𝜑 ↔ 𝜓)) | |
2 | simprim 164 | . . 3 ⊢ (¬ ((𝜑 ↔ 𝜓) → ¬ (𝜒 ↔ 𝜃)) → (𝜒 ↔ 𝜃)) | |
3 | 1, 2 | imbi12d 337 | . 2 ⊢ (¬ ((𝜑 ↔ 𝜓) → ¬ (𝜒 ↔ 𝜃)) → ((𝜑 → 𝜒) ↔ (𝜓 → 𝜃))) |
4 | 3 | expi 163 | 1 ⊢ ((𝜑 ↔ 𝜓) → ((𝜒 ↔ 𝜃) → ((𝜑 → 𝜒) ↔ (𝜓 → 𝜃)))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 |
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 199 |
This theorem is referenced by: imbi12i 343 bj-imbi12 33439 ifpbi12 39256 ifpbi13 39257 imbi13 40279 imbi13VD 40633 sbcssgVD 40642 |
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