Theorem orordi

Description: Distribution of disjunction over disjunction. (Contributed by NM, 25-Feb-1995)

	Ref	Expression
Assertion	orordi	⊢ ( ( 𝜑 ∨ ( 𝜓 ∨ 𝜒 ) ) ↔ ( ( 𝜑 ∨ 𝜓 ) ∨ ( 𝜑 ∨ 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		oridm	⊢ ( ( 𝜑 ∨ 𝜑 ) ↔ 𝜑 )
2	1	orbi1i	⊢ ( ( ( 𝜑 ∨ 𝜑 ) ∨ ( 𝜓 ∨ 𝜒 ) ) ↔ ( 𝜑 ∨ ( 𝜓 ∨ 𝜒 ) ) )
3		or4	⊢ ( ( ( 𝜑 ∨ 𝜑 ) ∨ ( 𝜓 ∨ 𝜒 ) ) ↔ ( ( 𝜑 ∨ 𝜓 ) ∨ ( 𝜑 ∨ 𝜒 ) ) )
4	2 3	bitr3i	⊢ ( ( 𝜑 ∨ ( 𝜓 ∨ 𝜒 ) ) ↔ ( ( 𝜑 ∨ 𝜓 ) ∨ ( 𝜑 ∨ 𝜒 ) ) )