Theorem anandir

Description: Distribution of conjunction over conjunction. (Contributed by NM, 24-Aug-1995)

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

Proof

Step	Hyp	Ref	Expression
1		anidm	⊢ ( ( 𝜒 ∧ 𝜒 ) ↔ 𝜒 )
2	1	anbi2i	⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∧ ( 𝜒 ∧ 𝜒 ) ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) )
3		an4	⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∧ ( 𝜒 ∧ 𝜒 ) ) ↔ ( ( 𝜑 ∧ 𝜒 ) ∧ ( 𝜓 ∧ 𝜒 ) ) )
4	2 3	bitr3i	⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) ↔ ( ( 𝜑 ∧ 𝜒 ) ∧ ( 𝜓 ∧ 𝜒 ) ) )