Theorem ecase2d

Description: Deduction for elimination by cases. (Contributed by NM, 21-Apr-1994) (Proof shortened by Wolf Lammen, 19-Sep-2024)

	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	1 2	mpnanrd	$$⊢\phi \to \neg \chi$$
6	1 3	mpnanrd	$$⊢\phi \to \neg \theta$$
7	4	ord	$$⊢\phi \to \left(\neg \tau \to \left(\chi \vee \theta \right)\right)$$
8	5 6 7	mtord	$$⊢\phi \to \neg \neg \tau$$
9	8	notnotrd	$$⊢\phi \to \tau$$