Theorem 3jao

Description: Disjunction of three antecedents. (Contributed by NM, 8-Apr-1994)

	Ref	Expression
Assertion	3jao	⊢ ( ( ( 𝜑 → 𝜓 ) ∧ ( 𝜒 → 𝜓 ) ∧ ( 𝜃 → 𝜓 ) ) → ( ( 𝜑 ∨ 𝜒 ∨ 𝜃 ) → 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		jao	⊢ ( ( 𝜑 → 𝜓 ) → ( ( 𝜒 → 𝜓 ) → ( ( 𝜑 ∨ 𝜒 ) → 𝜓 ) ) )
2		df-3or	⊢ ( ( 𝜑 ∨ 𝜒 ∨ 𝜃 ) ↔ ( ( 𝜑 ∨ 𝜒 ) ∨ 𝜃 ) )
3		jao	⊢ ( ( ( 𝜑 ∨ 𝜒 ) → 𝜓 ) → ( ( 𝜃 → 𝜓 ) → ( ( ( 𝜑 ∨ 𝜒 ) ∨ 𝜃 ) → 𝜓 ) ) )
4	2 3	syl7bi	⊢ ( ( ( 𝜑 ∨ 𝜒 ) → 𝜓 ) → ( ( 𝜃 → 𝜓 ) → ( ( 𝜑 ∨ 𝜒 ∨ 𝜃 ) → 𝜓 ) ) )
5	1 4	syl6	⊢ ( ( 𝜑 → 𝜓 ) → ( ( 𝜒 → 𝜓 ) → ( ( 𝜃 → 𝜓 ) → ( ( 𝜑 ∨ 𝜒 ∨ 𝜃 ) → 𝜓 ) ) ) )
6	5	3imp	⊢ ( ( ( 𝜑 → 𝜓 ) ∧ ( 𝜒 → 𝜓 ) ∧ ( 𝜃 → 𝜓 ) ) → ( ( 𝜑 ∨ 𝜒 ∨ 𝜃 ) → 𝜓 ) )