Theorem jaoian

Description: Inference disjoining the antecedents of two implications. (Contributed by NM, 23-Oct-2005)

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

Proof

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