Theorem jaodan

Description: Deduction disjoining the antecedents of two implications. (Contributed by NM, 14-Oct-2005)

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

Proof

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