Theorem orass

Description: Associative law for disjunction. Theorem *4.33 of WhiteheadRussell p. 118. (Contributed by NM, 5-Aug-1993) (Proof shortened by Andrew Salmon, 26-Jun-2011)

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

Proof

Step	Hyp	Ref	Expression
1		orcom	⊢ ( ( ( 𝜑 ∨ 𝜓 ) ∨ 𝜒 ) ↔ ( 𝜒 ∨ ( 𝜑 ∨ 𝜓 ) ) )
2		or12	⊢ ( ( 𝜒 ∨ ( 𝜑 ∨ 𝜓 ) ) ↔ ( 𝜑 ∨ ( 𝜒 ∨ 𝜓 ) ) )
3		orcom	⊢ ( ( 𝜒 ∨ 𝜓 ) ↔ ( 𝜓 ∨ 𝜒 ) )
4	3	orbi2i	⊢ ( ( 𝜑 ∨ ( 𝜒 ∨ 𝜓 ) ) ↔ ( 𝜑 ∨ ( 𝜓 ∨ 𝜒 ) ) )
5	1 2 4	3bitri	⊢ ( ( ( 𝜑 ∨ 𝜓 ) ∨ 𝜒 ) ↔ ( 𝜑 ∨ ( 𝜓 ∨ 𝜒 ) ) )