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| Description: Weak deduction theorem eliminating three hypotheses. See comments in dedth2h 4585. (Contributed by NM, 15-May-1999.) | 
| Ref | Expression | 
|---|---|
| dedth3h.1 | ⊢ (𝐴 = if(𝜑, 𝐴, 𝐷) → (𝜃 ↔ 𝜏)) | 
| dedth3h.2 | ⊢ (𝐵 = if(𝜓, 𝐵, 𝑅) → (𝜏 ↔ 𝜂)) | 
| dedth3h.3 | ⊢ (𝐶 = if(𝜒, 𝐶, 𝑆) → (𝜂 ↔ 𝜁)) | 
| dedth3h.4 | ⊢ 𝜁 | 
| Ref | Expression | 
|---|---|
| dedth3h | ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | dedth3h.1 | . . . 4 ⊢ (𝐴 = if(𝜑, 𝐴, 𝐷) → (𝜃 ↔ 𝜏)) | |
| 2 | 1 | imbi2d 340 | . . 3 ⊢ (𝐴 = if(𝜑, 𝐴, 𝐷) → (((𝜓 ∧ 𝜒) → 𝜃) ↔ ((𝜓 ∧ 𝜒) → 𝜏))) | 
| 3 | dedth3h.2 | . . . 4 ⊢ (𝐵 = if(𝜓, 𝐵, 𝑅) → (𝜏 ↔ 𝜂)) | |
| 4 | dedth3h.3 | . . . 4 ⊢ (𝐶 = if(𝜒, 𝐶, 𝑆) → (𝜂 ↔ 𝜁)) | |
| 5 | dedth3h.4 | . . . 4 ⊢ 𝜁 | |
| 6 | 3, 4, 5 | dedth2h 4585 | . . 3 ⊢ ((𝜓 ∧ 𝜒) → 𝜏) | 
| 7 | 2, 6 | dedth 4584 | . 2 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → 𝜃)) | 
| 8 | 7 | 3impib 1117 | 1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ifcif 4525 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-if 4526 | 
| This theorem is referenced by: dedth3v 4589 faclbnd4lem2 14333 dvdsle 16347 gcdaddm 16562 ipdiri 30849 hvaddcan 31089 hvsubadd 31096 norm3dif 31169 omlsii 31422 chjass 31552 ledi 31559 spansncv 31672 pjcjt2 31711 pjopyth 31739 hoaddass 31801 hocsubdir 31804 hoddi 32009 | 
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