Theorem anandi

Description: Distribution of conjunction over conjunction. (Contributed by NM, 14-Aug-1995)

	Ref	Expression
Assertion	anandi	⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝜒 ) ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∧ ( 𝜑 ∧ 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		anidm	⊢ ( ( 𝜑 ∧ 𝜑 ) ↔ 𝜑 )
2	1	anbi1i	⊢ ( ( ( 𝜑 ∧ 𝜑 ) ∧ ( 𝜓 ∧ 𝜒 ) ) ↔ ( 𝜑 ∧ ( 𝜓 ∧ 𝜒 ) ) )
3		an4	⊢ ( ( ( 𝜑 ∧ 𝜑 ) ∧ ( 𝜓 ∧ 𝜒 ) ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∧ ( 𝜑 ∧ 𝜒 ) ) )
4	2 3	bitr3i	⊢ ( ( 𝜑 ∧ ( 𝜓 ∧ 𝜒 ) ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∧ ( 𝜑 ∧ 𝜒 ) ) )