![]() |
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 | ancom 453 | . . 3 ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜓 ∧ 𝜑)) | |
2 | 1 | anbi1i 615 | . 2 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ ((𝜓 ∧ 𝜑) ∧ 𝜒)) |
3 | anass 461 | . 2 ⊢ (((𝜓 ∧ 𝜑) ∧ 𝜒) ↔ (𝜓 ∧ (𝜑 ∧ 𝜒))) | |
4 | 2, 3 | bitri 267 | 1 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ (𝜓 ∧ (𝜑 ∧ 𝜒))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ wa 387 |
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 df-an 388 |
This theorem is referenced by: an12 633 fncnv 6265 mpocurryd 7744 rexuz2 12119 logfac2 25510 ltgov 26100 brimg 32959 eldmqsres 35027 xrninxp2 35126 |
Copyright terms: Public domain | W3C validator |