Theorem ecase2d

Description: Deduction for elimination by cases. (Contributed by NM, 21-Apr-1994) (Proof shortened by Wolf Lammen, 22-Dec-2012)

	Ref	Expression
Hypotheses	ecase2d.1	$$⊢\phi \to \psi$$
	ecase2d.2	$$⊢\phi \to \neg \left(\psi \wedge \chi \right)$$
	ecase2d.3	$$⊢\phi \to \neg \left(\psi \wedge \theta \right)$$
	ecase2d.4	$$⊢\phi \to \left(\tau \vee \left(\chi \vee \theta \right)\right)$$
Assertion	ecase2d	$$⊢\phi \to \tau$$

Proof

Step	Hyp	Ref	Expression
1		ecase2d.1	$$⊢\phi \to \psi$$
2		ecase2d.2	$$⊢\phi \to \neg \left(\psi \wedge \chi \right)$$
3		ecase2d.3	$$⊢\phi \to \neg \left(\psi \wedge \theta \right)$$
4		ecase2d.4	$$⊢\phi \to \left(\tau \vee \left(\chi \vee \theta \right)\right)$$
5		idd	$$⊢\phi \to \left(\tau \to \tau \right)$$
6	2	pm2.21d	$$⊢\phi \to \left(\left(\psi \wedge \chi \right)\to \tau \right)$$
7	1 6	mpand	$$⊢\phi \to \left(\chi \to \tau \right)$$
8	3	pm2.21d	$$⊢\phi \to \left(\left(\psi \wedge \theta \right)\to \tau \right)$$
9	1 8	mpand	$$⊢\phi \to \left(\theta \to \tau \right)$$
10	7 9	jaod	$$⊢\phi \to \left(\left(\chi \vee \theta \right)\to \tau \right)$$
11	5 10 4	mpjaod	$$⊢\phi \to \tau$$