Theorem 3jaodan

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

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

Proof

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