Theorem ecase13d

Description: Deduction for elimination by cases. (Contributed by Jeff Hankins, 18-Aug-2009)

	Ref	Expression
Hypotheses	ecase13d.1	$$⊢\phi \to \neg \chi$$
	ecase13d.2	$$⊢\phi \to \neg \theta$$
	ecase13d.3	$$⊢\phi \to \left(\chi \vee \psi \vee \theta \right)$$
Assertion	ecase13d	$$⊢\phi \to \psi$$

Proof

Step	Hyp	Ref	Expression
1		ecase13d.1	$$⊢\phi \to \neg \chi$$
2		ecase13d.2	$$⊢\phi \to \neg \theta$$
3		ecase13d.3	$$⊢\phi \to \left(\chi \vee \psi \vee \theta \right)$$
4		3orass	$$⊢\left(\chi \vee \psi \vee \theta \right)\leftrightarrow \left(\chi \vee \left(\psi \vee \theta \right)\right)$$
5		df-or	$$⊢\left(\chi \vee \left(\psi \vee \theta \right)\right)\leftrightarrow \left(\neg \chi \to \left(\psi \vee \theta \right)\right)$$
6	4 5	bitri	$$⊢\left(\chi \vee \psi \vee \theta \right)\leftrightarrow \left(\neg \chi \to \left(\psi \vee \theta \right)\right)$$
7	3 6	sylib	$$⊢\phi \to \left(\neg \chi \to \left(\psi \vee \theta \right)\right)$$
8	1 7	mpd	$$⊢\phi \to \left(\psi \vee \theta \right)$$
9		orcom	$$⊢\left(\psi \vee \theta \right)\leftrightarrow \left(\theta \vee \psi \right)$$
10		df-or	$$⊢\left(\theta \vee \psi \right)\leftrightarrow \left(\neg \theta \to \psi \right)$$
11	9 10	bitri	$$⊢\left(\psi \vee \theta \right)\leftrightarrow \left(\neg \theta \to \psi \right)$$
12	8 11	sylib	$$⊢\phi \to \left(\neg \theta \to \psi \right)$$
13	2 12	mpd	$$⊢\phi \to \psi$$