|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > 3an4anass | Structured version Visualization version GIF version | ||
| Description: Associative law for four conjunctions with a triple conjunction. (Contributed by Alexander van der Vekens, 24-Jun-2018.) | 
| Ref | Expression | 
|---|---|
| 3an4anass | ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-3an 1089 | . . 3 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜒)) | |
| 2 | 1 | anbi1i 624 | . 2 ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) ↔ (((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃)) | 
| 3 | anass 468 | . 2 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃))) | |
| 4 | 2, 3 | bitri 275 | 1 ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 ∧ w3a 1087 | 
| 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 df-3an 1089 | 
| This theorem is referenced by: oeeui 8640 isclwlkupgr 29798 clwlkclwwlk 30021 13an22anass 32483 bnj557 34915 cantnf2 43338 3an4ancom24 47281 isthincd2 49086 | 
| Copyright terms: Public domain | W3C validator |