Theorem 3an4ancom24

Description: Commutative law for a conjunction with a triple conjunction: second and forth positions interchanged. (Contributed by AV, 18-Feb-2022)

	Ref	Expression
Assertion	3an4ancom24	⊢ ( ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) ∧ 𝜃 ) ↔ ( ( 𝜑 ∧ 𝜃 ∧ 𝜒 ) ∧ 𝜓 ) )

Proof

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