Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  3an4anass Structured version   Visualization version   GIF version

Theorem 3an4anass 1133
 Description: Associative law for four conjunctions with a triple conjunction. (Contributed by Alexander van der Vekens, 24-Jun-2018.)
Assertion
Ref Expression
3an4anass (((𝜑𝜓𝜒) ∧ 𝜃) ↔ ((𝜑𝜓) ∧ (𝜒𝜃)))

Proof of Theorem 3an4anass
StepHypRef Expression
1 df-3an 1113 . . 3 ((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∧ 𝜒))
21anbi1i 617 . 2 (((𝜑𝜓𝜒) ∧ 𝜃) ↔ (((𝜑𝜓) ∧ 𝜒) ∧ 𝜃))
3 anass 462 . 2 ((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ↔ ((𝜑𝜓) ∧ (𝜒𝜃)))
42, 3bitri 267 1 (((𝜑𝜓𝜒) ∧ 𝜃) ↔ ((𝜑𝜓) ∧ (𝜒𝜃)))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 198   ∧ wa 386   ∧ w3a 1111 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 387  df-3an 1113 This theorem is referenced by:  oeeui  7954  isclwlkupgr  27087  clwlkclwwlk  27338  clwlkclwwlkOLD  27339  bnj557  31513  3an4ancom24  42169
 Copyright terms: Public domain W3C validator