|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > an21 | Structured version Visualization version GIF version | ||
| Description: Swap two conjuncts. (Contributed by Peter Mazsa, 18-Sep-2022.) | 
| Ref | Expression | 
|---|---|
| an21 | ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ (𝜓 ∧ (𝜑 ∧ 𝜒))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | biid 261 | . . 3 ⊢ ((𝜑 ∧ 𝜒) ↔ (𝜑 ∧ 𝜒)) | |
| 2 | 1 | bianassc 643 | . 2 ⊢ ((𝜓 ∧ (𝜑 ∧ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜒)) | 
| 3 | 2 | bicomi 224 | 1 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ (𝜓 ∧ (𝜑 ∧ 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: an32 646 an13 647 indifdi 4293 fncnv 6638 mpocurryd 8295 rexuz2 12942 resmndismnd 18822 imasabl 19895 logfac2 27262 ltgov 28606 brimg 35939 eldmqsres 38289 xrninxp2 38395 | 
| Copyright terms: Public domain | W3C validator |