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Theorem 3an4anass 1104
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 1088 . . 3 ((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∧ 𝜒))
21anbi1i 624 . 2 (((𝜑𝜓𝜒) ∧ 𝜃) ↔ (((𝜑𝜓) ∧ 𝜒) ∧ 𝜃))
3 anass 468 . 2 ((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ↔ ((𝜑𝜓) ∧ (𝜒𝜃)))
42, 3bitri 275 1 (((𝜑𝜓𝜒) ∧ 𝜃) ↔ ((𝜑𝜓) ∧ (𝜒𝜃)))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  w3a 1086
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  df-3an 1088
This theorem is referenced by:  oeeui  8639  isclwlkupgr  29811  clwlkclwwlk  30031  13an22anass  32493  bnj557  34894  cantnf2  43315  3an4ancom24  47219  isthincd2  48838
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