Theorem 3ornot23

Description: If the second and third disjuncts of a true triple disjunction are false, then the first disjunct is true. Automatically derived from 3ornot23VD . (Contributed by Alan Sare, 31-Dec-2011) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	3ornot23	$$⊢\left(\neg \phi \wedge \neg \psi \right)\to \left(\left(\chi \vee \phi \vee \psi \right)\to \chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		idd	$$⊢\neg \phi \to \left(\chi \to \chi \right)$$
2		pm2.21	$$⊢\neg \phi \to \left(\phi \to \chi \right)$$
3		pm2.21	$$⊢\neg \psi \to \left(\psi \to \chi \right)$$
4	1 2 3	3jaao	$$⊢\left(\neg \phi \wedge \neg \phi \wedge \neg \psi \right)\to \left(\left(\chi \vee \phi \vee \psi \right)\to \chi \right)$$
5	4	3anidm12	$$⊢\left(\neg \phi \wedge \neg \psi \right)\to \left(\left(\chi \vee \phi \vee \psi \right)\to \chi \right)$$