Theorem 3jaoian

Description: Disjunction of three antecedents (inference). (Contributed by NM, 14-Oct-2005)

	Ref	Expression
Hypotheses	3jaoian.1	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜒 )
	3jaoian.2	⊢ ( ( 𝜃 ∧ 𝜓 ) → 𝜒 )
	3jaoian.3	⊢ ( ( 𝜏 ∧ 𝜓 ) → 𝜒 )
Assertion	3jaoian	⊢ ( ( ( 𝜑 ∨ 𝜃 ∨ 𝜏 ) ∧ 𝜓 ) → 𝜒 )

Proof

Step	Hyp	Ref	Expression
1		3jaoian.1	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜒 )
2		3jaoian.2	⊢ ( ( 𝜃 ∧ 𝜓 ) → 𝜒 )
3		3jaoian.3	⊢ ( ( 𝜏 ∧ 𝜓 ) → 𝜒 )
4	1	ex	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
5	2	ex	⊢ ( 𝜃 → ( 𝜓 → 𝜒 ) )
6	3	ex	⊢ ( 𝜏 → ( 𝜓 → 𝜒 ) )
7	4 5 6	3jaoi	⊢ ( ( 𝜑 ∨ 𝜃 ∨ 𝜏 ) → ( 𝜓 → 𝜒 ) )
8	7	imp	⊢ ( ( ( 𝜑 ∨ 𝜃 ∨ 𝜏 ) ∧ 𝜓 ) → 𝜒 )