Theorem ecased

Description: Deduction for elimination by cases. (Contributed by NM, 8-Oct-2012)

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

Proof

Step	Hyp	Ref	Expression
1		ecased.1	$$⊢\phi \to \left(\neg \psi \to \theta \right)$$
2		ecased.2	$$⊢\phi \to \left(\neg \chi \to \theta \right)$$
3		ecased.3	$$⊢\phi \to \left(\left(\psi \wedge \chi \right)\to \theta \right)$$
4		pm3.11	$$⊢\neg \left(\neg \psi \vee \neg \chi \right)\to \left(\psi \wedge \chi \right)$$
5	4 3	syl5	$$⊢\phi \to \left(\neg \left(\neg \psi \vee \neg \chi \right)\to \theta \right)$$
6	1 2 5	ecase3d	$$⊢\phi \to \theta$$