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| Description: Inference separating a disjunct of an antecedent. (Contributed by Alexander van der Vekens, 25-May-2018.) | 
| Ref | Expression | 
|---|---|
| jaoi3.1 | ⊢ (𝜑 → 𝜓) | 
| jaoi3.2 | ⊢ ((¬ 𝜑 ∧ 𝜒) → 𝜓) | 
| Ref | Expression | 
|---|---|
| jaoi3 | ⊢ ((𝜑 ∨ 𝜒) → 𝜓) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | jaoi3.1 | . . 3 ⊢ (𝜑 → 𝜓) | |
| 2 | jaoi3.2 | . . 3 ⊢ ((¬ 𝜑 ∧ 𝜒) → 𝜓) | |
| 3 | 1, 2 | jaoi 858 | . 2 ⊢ ((𝜑 ∨ (¬ 𝜑 ∧ 𝜒)) → 𝜓) | 
| 4 | 3 | jaoi2 1060 | 1 ⊢ ((𝜑 ∨ 𝜒) → 𝜓) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 848 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 | 
| This theorem is referenced by: 2mpo0 7682 bropopvvv 8115 bropfvvvv 8117 ssnn0fi 14026 swrdnd 14692 swrdnnn0nd 14694 swrdnd0 14695 pfxnd0 14726 line2ylem 48672 line2xlem 48674 itsclc0xyqsol 48689 | 
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