Theorem andir

Description: Distributive law for conjunction. (Contributed by NM, 12-Aug-1994)

	Ref	Expression
Assertion	andir	⊢ ( ( ( 𝜑 ∨ 𝜓 ) ∧ 𝜒 ) ↔ ( ( 𝜑 ∧ 𝜒 ) ∨ ( 𝜓 ∧ 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		andi	⊢ ( ( 𝜒 ∧ ( 𝜑 ∨ 𝜓 ) ) ↔ ( ( 𝜒 ∧ 𝜑 ) ∨ ( 𝜒 ∧ 𝜓 ) ) )
2		ancom	⊢ ( ( ( 𝜑 ∨ 𝜓 ) ∧ 𝜒 ) ↔ ( 𝜒 ∧ ( 𝜑 ∨ 𝜓 ) ) )
3		ancom	⊢ ( ( 𝜑 ∧ 𝜒 ) ↔ ( 𝜒 ∧ 𝜑 ) )
4		ancom	⊢ ( ( 𝜓 ∧ 𝜒 ) ↔ ( 𝜒 ∧ 𝜓 ) )
5	3 4	orbi12i	⊢ ( ( ( 𝜑 ∧ 𝜒 ) ∨ ( 𝜓 ∧ 𝜒 ) ) ↔ ( ( 𝜒 ∧ 𝜑 ) ∨ ( 𝜒 ∧ 𝜓 ) ) )
6	1 2 5	3bitr4i	⊢ ( ( ( 𝜑 ∨ 𝜓 ) ∧ 𝜒 ) ↔ ( ( 𝜑 ∧ 𝜒 ) ∨ ( 𝜓 ∧ 𝜒 ) ) )