Theorem 3jaoi

Description: Disjunction of three antecedents (inference). (Contributed by NM, 12-Sep-1995) (Proof shortened by Garrett Katz, 16-Jun-2026)

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

Proof

Step	Hyp	Ref	Expression
1		3jaoi.1	$$⊢\phi \to \psi$$
2		3jaoi.2	$$⊢\chi \to \psi$$
3		3jaoi.3	$$⊢\theta \to \psi$$
4		3jaob	$$⊢\left(\left(\phi \vee \chi \vee \theta \right)\to \psi \right)\leftrightarrow \left(\left(\phi \to \psi \right)\wedge \left(\chi \to \psi \right)\wedge \left(\theta \to \psi \right)\right)$$
5	1 2 3 4	mpbir3an	$$⊢\left(\phi \vee \chi \vee \theta \right)\to \psi$$