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| Description: Distribution of conjunction over conjunction. (Contributed by NM, 24-Aug-1995.) | 
| Ref | Expression | 
|---|---|
| anandir | ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ ((𝜑 ∧ 𝜒) ∧ (𝜓 ∧ 𝜒))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | anidm 564 | . . 3 ⊢ ((𝜒 ∧ 𝜒) ↔ 𝜒) | |
| 2 | 1 | anbi2i 623 | . 2 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜒)) | 
| 3 | an4 656 | . 2 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜒)) ↔ ((𝜑 ∧ 𝜒) ∧ (𝜓 ∧ 𝜒))) | |
| 4 | 2, 3 | bitr3i 277 | 1 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ ((𝜑 ∧ 𝜒) ∧ (𝜓 ∧ 𝜒))) | 
| 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: anandi3r 1103 disjxun 5141 fununi 6641 imadif 6650 elfzuzb 13558 frgr3v 30294 5oalem3 31675 5oalem5 31677 refrelredund4 38636 nzin 44337 un2122 44810 | 
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