Theorem or3dir

Description: Distributive law for disjunction. (Contributed by Thierry Arnoux, 3-Jul-2017)

	Ref	Expression
Assertion	or3dir	⊢ ( ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) ∨ 𝜏 ) ↔ ( ( 𝜑 ∨ 𝜏 ) ∧ ( 𝜓 ∨ 𝜏 ) ∧ ( 𝜒 ∨ 𝜏 ) ) )

Proof

Step	Hyp	Ref	Expression
1		or3di	⊢ ( ( 𝜏 ∨ ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) ) ↔ ( ( 𝜏 ∨ 𝜑 ) ∧ ( 𝜏 ∨ 𝜓 ) ∧ ( 𝜏 ∨ 𝜒 ) ) )
2		orcom	⊢ ( ( 𝜏 ∨ ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) ) ↔ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) ∨ 𝜏 ) )
3		orcom	⊢ ( ( 𝜏 ∨ 𝜑 ) ↔ ( 𝜑 ∨ 𝜏 ) )
4		orcom	⊢ ( ( 𝜏 ∨ 𝜓 ) ↔ ( 𝜓 ∨ 𝜏 ) )
5		orcom	⊢ ( ( 𝜏 ∨ 𝜒 ) ↔ ( 𝜒 ∨ 𝜏 ) )
6	3 4 5	3anbi123i	⊢ ( ( ( 𝜏 ∨ 𝜑 ) ∧ ( 𝜏 ∨ 𝜓 ) ∧ ( 𝜏 ∨ 𝜒 ) ) ↔ ( ( 𝜑 ∨ 𝜏 ) ∧ ( 𝜓 ∨ 𝜏 ) ∧ ( 𝜒 ∨ 𝜏 ) ) )
7	1 2 6	3bitr3i	⊢ ( ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) ∨ 𝜏 ) ↔ ( ( 𝜑 ∨ 𝜏 ) ∧ ( 𝜓 ∨ 𝜏 ) ∧ ( 𝜒 ∨ 𝜏 ) ) )