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| Description: Disjunction of three antecedents. (Contributed by NM, 13-Sep-2011.) (Proof shortened by Hongxiu Chen, 29-Jun-2025.) | 
| Ref | Expression | 
|---|---|
| 3jaob | ⊢ (((𝜑 ∨ 𝜒 ∨ 𝜃) → 𝜓) ↔ ((𝜑 → 𝜓) ∧ (𝜒 → 𝜓) ∧ (𝜃 → 𝜓))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | pm5.53 1006 | . 2 ⊢ ((((𝜑 ∨ 𝜒) ∨ 𝜃) → 𝜓) ↔ (((𝜑 → 𝜓) ∧ (𝜒 → 𝜓)) ∧ (𝜃 → 𝜓))) | |
| 2 | df-3or 1087 | . . 3 ⊢ ((𝜑 ∨ 𝜒 ∨ 𝜃) ↔ ((𝜑 ∨ 𝜒) ∨ 𝜃)) | |
| 3 | 2 | imbi1i 349 | . 2 ⊢ (((𝜑 ∨ 𝜒 ∨ 𝜃) → 𝜓) ↔ (((𝜑 ∨ 𝜒) ∨ 𝜃) → 𝜓)) | 
| 4 | df-3an 1088 | . 2 ⊢ (((𝜑 → 𝜓) ∧ (𝜒 → 𝜓) ∧ (𝜃 → 𝜓)) ↔ (((𝜑 → 𝜓) ∧ (𝜒 → 𝜓)) ∧ (𝜃 → 𝜓))) | |
| 5 | 1, 3, 4 | 3bitr4i 303 | 1 ⊢ (((𝜑 ∨ 𝜒 ∨ 𝜃) → 𝜓) ↔ ((𝜑 → 𝜓) ∧ (𝜒 → 𝜓) ∧ (𝜃 → 𝜓))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 ∨ w3o 1085 ∧ w3a 1086 | 
| 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 848 df-3or 1087 df-3an 1088 | 
| This theorem is referenced by: (None) | 
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