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Theorem 3an4anass 1099
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 1083 . . 3 ((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∧ 𝜒))
21anbi1i 625 . 2 (((𝜑𝜓𝜒) ∧ 𝜃) ↔ (((𝜑𝜓) ∧ 𝜒) ∧ 𝜃))
3 anass 471 . 2 ((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ↔ ((𝜑𝜓) ∧ (𝜒𝜃)))
42, 3bitri 277 1 (((𝜑𝜓𝜒) ∧ 𝜃) ↔ ((𝜑𝜓) ∧ (𝜒𝜃)))
Colors of variables: wff setvar class
Syntax hints:  wb 208  wa 398  w3a 1081
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 209  df-an 399  df-3an 1083
This theorem is referenced by:  oeeui  8220  isclwlkupgr  27551  clwlkclwwlk  27772  bnj557  32161  3an4ancom24  43453
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