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	$$⊢\left(\left(\phi \vee \psi \right)\vee \chi \right)\leftrightarrow \left(\phi \vee \left(\psi \vee \chi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		orcom	$$⊢\left(\left(\phi \vee \psi \right)\vee \chi \right)\leftrightarrow \left(\chi \vee \left(\phi \vee \psi \right)\right)$$
2		or12	$$⊢\left(\chi \vee \left(\phi \vee \psi \right)\right)\leftrightarrow \left(\phi \vee \left(\chi \vee \psi \right)\right)$$
3		orcom	$$⊢\left(\chi \vee \psi \right)\leftrightarrow \left(\psi \vee \chi \right)$$
4	3	orbi2i	$$⊢\left(\phi \vee \left(\chi \vee \psi \right)\right)\leftrightarrow \left(\phi \vee \left(\psi \vee \chi \right)\right)$$
5	1 2 4	3bitri	$$⊢\left(\left(\phi \vee \psi \right)\vee \chi \right)\leftrightarrow \left(\phi \vee \left(\psi \vee \chi \right)\right)$$