Theorem jaoi3

Description: Inference separating a disjunct of an antecedent. (Contributed by Alexander van der Vekens, 25-May-2018)

	Ref	Expression
Hypotheses	jaoi3.1	⊢ ( 𝜑 → 𝜓 )
	jaoi3.2	⊢ ( ( ¬ 𝜑 ∧ 𝜒 ) → 𝜓 )
Assertion	jaoi3	⊢ ( ( 𝜑 ∨ 𝜒 ) → 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		jaoi3.1	⊢ ( 𝜑 → 𝜓 )
2		jaoi3.2	⊢ ( ( ¬ 𝜑 ∧ 𝜒 ) → 𝜓 )
3	1 2	jaoi	⊢ ( ( 𝜑 ∨ ( ¬ 𝜑 ∧ 𝜒 ) ) → 𝜓 )
4	3	jaoi2	⊢ ( ( 𝜑 ∨ 𝜒 ) → 𝜓 )