Theorem ecase3ad

Description: Deduction for elimination by cases. (Contributed by NM, 24-May-2013)

	Ref	Expression
Hypotheses	ecase3ad.1	$$⊢\phi \to \left(\psi \to \theta \right)$$
	ecase3ad.2	$$⊢\phi \to \left(\chi \to \theta \right)$$
	ecase3ad.3	$$⊢\phi \to \left(\left(\neg \psi \wedge \neg \chi \right)\to \theta \right)$$
Assertion	ecase3ad	$$⊢\phi \to \theta$$

Proof

Step	Hyp	Ref	Expression
1		ecase3ad.1	$$⊢\phi \to \left(\psi \to \theta \right)$$
2		ecase3ad.2	$$⊢\phi \to \left(\chi \to \theta \right)$$
3		ecase3ad.3	$$⊢\phi \to \left(\left(\neg \psi \wedge \neg \chi \right)\to \theta \right)$$
4		notnotr	$$⊢\neg \neg \psi \to \psi$$
5	4 1	syl5	$$⊢\phi \to \left(\neg \neg \psi \to \theta \right)$$
6		notnotr	$$⊢\neg \neg \chi \to \chi$$
7	6 2	syl5	$$⊢\phi \to \left(\neg \neg \chi \to \theta \right)$$
8	5 7 3	ecased	$$⊢\phi \to \theta$$