Theorem 3an4ancom24 43475

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	⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) ↔ ((𝜑 ∧ 𝜃 ∧ 𝜒) ∧ 𝜓))

Proof of Theorem 3an4ancom24

Step	Hyp	Ref	Expression
1		an4com24 43474	. 2 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜃) ∧ (𝜒 ∧ 𝜓)))
2		3an4anass 1101	. 2 ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)))
3		3an4anass 1101	. 2 ⊢ (((𝜑 ∧ 𝜃 ∧ 𝜒) ∧ 𝜓) ↔ ((𝜑 ∧ 𝜃) ∧ (𝜒 ∧ 𝜓)))
4	1, 2, 3	3bitr4i 305	1 ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) ↔ ((𝜑 ∧ 𝜃 ∧ 𝜒) ∧ 𝜓))

Syntax hints:   ↔ wb 208   ∧ wa 398   ∧ w3a 1083

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 1085

This theorem is referenced by: (None)