Theorem or32

Description: A rearrangement of disjuncts. (Contributed by NM, 18-Oct-1995) (Proof shortened by Andrew Salmon, 26-Jun-2011)

	Ref	Expression
Assertion	or32	$$⊢\left(\left(\phi \vee \psi \right)\vee \chi \right)\leftrightarrow \left(\left(\phi \vee \chi \right)\vee \psi \right)$$

Proof

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