Theorem 3an4anass 1105

Description: Associative law for four conjunctions with a triple conjunction. (Contributed by Alexander van der Vekens, 24-Jun-2018.)

Assertion

Ref	Expression
3an4anass	⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)))

Proof of Theorem 3an4anass

Step	Hyp	Ref	Expression
1		df-3an 1089	. . 3 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜒))
2	1	anbi1i 624	. 2 ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) ↔ (((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃))
3		anass 468	. 2 ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)))
4	2, 3	bitri 275	1 ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1087

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 1089

This theorem is referenced by:  oeeui  8640  isclwlkupgr  29798  clwlkclwwlk  30021  13an22anass  32483  bnj557  34915  cantnf2  43338  3an4ancom24  47281  isthincd2  49086