Theorem an4

Description: Rearrangement of 4 conjuncts. (Contributed by NM, 10-Jul-1994)

	Ref	Expression
Assertion	an4	⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∧ ( 𝜒 ∧ 𝜃 ) ) ↔ ( ( 𝜑 ∧ 𝜒 ) ∧ ( 𝜓 ∧ 𝜃 ) ) )

Proof

Step	Hyp	Ref	Expression
1		anass	⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∧ ( 𝜒 ∧ 𝜃 ) ) ↔ ( 𝜑 ∧ ( 𝜓 ∧ ( 𝜒 ∧ 𝜃 ) ) ) )
2		an12	⊢ ( ( 𝜓 ∧ ( 𝜒 ∧ 𝜃 ) ) ↔ ( 𝜒 ∧ ( 𝜓 ∧ 𝜃 ) ) )
3	2	bianass	⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ ( 𝜒 ∧ 𝜃 ) ) ) ↔ ( ( 𝜑 ∧ 𝜒 ) ∧ ( 𝜓 ∧ 𝜃 ) ) )
4	1 3	bitri	⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∧ ( 𝜒 ∧ 𝜃 ) ) ↔ ( ( 𝜑 ∧ 𝜒 ) ∧ ( 𝜓 ∧ 𝜃 ) ) )