Theorem ecase3ad

Description: Deduction for elimination by cases. (Contributed by NM, 24-May-2013) (Proof shortened by Wolf Lammen, 20-Sep-2024)

	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	1	imp	$$⊢\left(\phi \wedge \psi \right)\to \theta$$
5	2	imp	$$⊢\left(\phi \wedge \chi \right)\to \theta$$
6	3	imp	$$⊢\left(\phi \wedge \left(\neg \psi \wedge \neg \chi \right)\right)\to \theta$$
7	4 5 6	pm2.61ddan	$$⊢\phi \to \theta$$