Theorem an42

Description: Rearrangement of 4 conjuncts. (Contributed by NM, 7-Feb-1996)

	Ref	Expression
Assertion	an42	⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∧ ( 𝜒 ∧ 𝜃 ) ) ↔ ( ( 𝜑 ∧ 𝜒 ) ∧ ( 𝜃 ∧ 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		an4	⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∧ ( 𝜒 ∧ 𝜃 ) ) ↔ ( ( 𝜑 ∧ 𝜒 ) ∧ ( 𝜓 ∧ 𝜃 ) ) )
2		ancom	⊢ ( ( 𝜓 ∧ 𝜃 ) ↔ ( 𝜃 ∧ 𝜓 ) )
3	2	anbi2i	⊢ ( ( ( 𝜑 ∧ 𝜒 ) ∧ ( 𝜓 ∧ 𝜃 ) ) ↔ ( ( 𝜑 ∧ 𝜒 ) ∧ ( 𝜃 ∧ 𝜓 ) ) )
4	1 3	bitri	⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∧ ( 𝜒 ∧ 𝜃 ) ) ↔ ( ( 𝜑 ∧ 𝜒 ) ∧ ( 𝜃 ∧ 𝜓 ) ) )