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Metamath Proof Explorer

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 )