Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  3an4ancom24 Structured version   Visualization version   GIF version

Theorem 3an4ancom24 42292
 Description: Commutative law for a conjunction with a triple conjunction: second and forth positions interchanged. (Contributed by AV, 18-Feb-2022.)
Assertion
Ref Expression
3an4ancom24 (((𝜑𝜓𝜒) ∧ 𝜃) ↔ ((𝜑𝜃𝜒) ∧ 𝜓))

Proof of Theorem 3an4ancom24
StepHypRef Expression
1 an4com24 42291 . 2 (((𝜑𝜓) ∧ (𝜒𝜃)) ↔ ((𝜑𝜃) ∧ (𝜒𝜓)))
2 3an4anass 1092 . 2 (((𝜑𝜓𝜒) ∧ 𝜃) ↔ ((𝜑𝜓) ∧ (𝜒𝜃)))
3 3an4anass 1092 . 2 (((𝜑𝜃𝜒) ∧ 𝜓) ↔ ((𝜑𝜃) ∧ (𝜒𝜓)))
41, 2, 33bitr4i 295 1 (((𝜑𝜓𝜒) ∧ 𝜃) ↔ ((𝜑𝜃𝜒) ∧ 𝜓))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 198   ∧ wa 386   ∧ w3a 1071 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 1073 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator