Theorem pm2.82

Description: Theorem *2.82 of WhiteheadRussell p. 108. (Contributed by NM, 3-Jan-2005)

	Ref	Expression
Assertion	pm2.82	$$⊢\left(\left(\phi \vee \psi \right)\vee \chi \right)\to \left(\left(\left(\phi \vee \neg \chi \right)\vee \theta \right)\to \left(\left(\phi \vee \psi \right)\vee \theta \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		pm2.24	$$⊢\chi \to \left(\neg \chi \to \psi \right)$$
2	1	orim2d	$$⊢\chi \to \left(\left(\phi \vee \neg \chi \right)\to \left(\phi \vee \psi \right)\right)$$
3	2	jao1i	$$⊢\left(\left(\phi \vee \psi \right)\vee \chi \right)\to \left(\left(\phi \vee \neg \chi \right)\to \left(\phi \vee \psi \right)\right)$$
4	3	orim1d	$$⊢\left(\left(\phi \vee \psi \right)\vee \chi \right)\to \left(\left(\left(\phi \vee \neg \chi \right)\vee \theta \right)\to \left(\left(\phi \vee \psi \right)\vee \theta \right)\right)$$