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	⊢ ( 𝜑 → 𝜓 )
	3jaoi.2	⊢ ( 𝜒 → 𝜓 )
	3jaoi.3	⊢ ( 𝜃 → 𝜓 )
Assertion	3jaoi	⊢ ( ( 𝜑 ∨ 𝜒 ∨ 𝜃 ) → 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		3jaoi.1	⊢ ( 𝜑 → 𝜓 )
2		3jaoi.2	⊢ ( 𝜒 → 𝜓 )
3		3jaoi.3	⊢ ( 𝜃 → 𝜓 )
4		3jaob	⊢ ( ( ( 𝜑 ∨ 𝜒 ∨ 𝜃 ) → 𝜓 ) ↔ ( ( 𝜑 → 𝜓 ) ∧ ( 𝜒 → 𝜓 ) ∧ ( 𝜃 → 𝜓 ) ) )
5	1 2 3 4	mpbir3an	⊢ ( ( 𝜑 ∨ 𝜒 ∨ 𝜃 ) → 𝜓 )