Theorem mpjao3dan

Description: Eliminate a three-way disjunction in a deduction. (Contributed by Thierry Arnoux, 13-Apr-2018) (Proof shortened by Wolf Lammen, 20-Apr-2024)

	Ref	Expression
Hypotheses	mpjao3dan.1	$$⊢\left(\phi \wedge \psi \right)\to \chi$$
	mpjao3dan.2	$$⊢\left(\phi \wedge \theta \right)\to \chi$$
	mpjao3dan.3	$$⊢\left(\phi \wedge \tau \right)\to \chi$$
	mpjao3dan.4	$$⊢\phi \to \left(\psi \vee \theta \vee \tau \right)$$
Assertion	mpjao3dan	$$⊢\phi \to \chi$$

Proof

Step	Hyp	Ref	Expression
1		mpjao3dan.1	$$⊢\left(\phi \wedge \psi \right)\to \chi$$
2		mpjao3dan.2	$$⊢\left(\phi \wedge \theta \right)\to \chi$$
3		mpjao3dan.3	$$⊢\left(\phi \wedge \tau \right)\to \chi$$
4		mpjao3dan.4	$$⊢\phi \to \left(\psi \vee \theta \vee \tau \right)$$
5	1 2 3	3jaodan	$$⊢\left(\phi \wedge \left(\psi \vee \theta \vee \tau \right)\right)\to \chi$$
6	4 5	mpdan	$$⊢\phi \to \chi$$