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| Mirrors > Home > MPE Home > Th. List > an42 | Structured version Visualization version GIF version | ||
| Description: Rearrangement of 4 conjuncts. (Contributed by NM, 7-Feb-1996.) |
| Ref | Expression |
|---|---|
| an42 | ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜒) ∧ (𝜃 ∧ 𝜓))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | an4 656 | . 2 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜒) ∧ (𝜓 ∧ 𝜃))) | |
| 2 | ancom 460 | . . 3 ⊢ ((𝜓 ∧ 𝜃) ↔ (𝜃 ∧ 𝜓)) | |
| 3 | 2 | anbi2i 623 | . 2 ⊢ (((𝜑 ∧ 𝜒) ∧ (𝜓 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜒) ∧ (𝜃 ∧ 𝜓))) |
| 4 | 1, 3 | bitri 275 | 1 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜒) ∧ (𝜃 ∧ 𝜓))) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 |
| 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 207 df-an 396 |
| This theorem is referenced by: an43 658 an33rean 1485 brecop2 8851 supmo 9492 infmo 9535 aceq1 10157 dfiso2 17816 eulerpartlemt0 34371 isbasisrelowllem1 37356 isbasisrelowllem2 37357 ifp1bi 43515 |
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