Theorem 13an22anass 1379

Description: Associative law for four conjunctions with a triple conjunction. (Contributed by Thierry Arnoux, 21-Jan-2025.) (Proof shortened by Wolf Lammen, 31-Oct-2026.)

Assertion

Ref	Expression
13an22anass	⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)))

Proof of Theorem 13an22anass

Step	Hyp	Ref	Expression
1		3anass 1111	. 2 ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (𝜓 ∧ (𝜒 ∧ 𝜃)))
2	1	bianass 655	1 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∧ w3a 1103

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105

This theorem is used by:  3anasss  1381  mpt3eqdv  7710  ressply1mon1p  34170  fwfffunres  35788