Theorem ordir

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

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

Proof

Step	Hyp	Ref	Expression
1		ordi	⊢ ( ( 𝜒 ∨ ( 𝜑 ∧ 𝜓 ) ) ↔ ( ( 𝜒 ∨ 𝜑 ) ∧ ( 𝜒 ∨ 𝜓 ) ) )
2		orcom	⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∨ 𝜒 ) ↔ ( 𝜒 ∨ ( 𝜑 ∧ 𝜓 ) ) )
3		orcom	⊢ ( ( 𝜑 ∨ 𝜒 ) ↔ ( 𝜒 ∨ 𝜑 ) )
4		orcom	⊢ ( ( 𝜓 ∨ 𝜒 ) ↔ ( 𝜒 ∨ 𝜓 ) )
5	3 4	anbi12i	⊢ ( ( ( 𝜑 ∨ 𝜒 ) ∧ ( 𝜓 ∨ 𝜒 ) ) ↔ ( ( 𝜒 ∨ 𝜑 ) ∧ ( 𝜒 ∨ 𝜓 ) ) )
6	1 2 5	3bitr4i	⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∨ 𝜒 ) ↔ ( ( 𝜑 ∨ 𝜒 ) ∧ ( 𝜓 ∨ 𝜒 ) ) )