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Mirrors > Home > MPE Home > Th. List > dedth3h | Structured version Visualization version GIF version |
Description: Weak deduction theorem eliminating three hypotheses. See comments in dedth2h 4590. (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 4590 | . . 3 ⊢ ((𝜓 ∧ 𝜒) → 𝜏) |
7 | 2, 6 | dedth 4589 | . 2 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → 𝜃)) |
8 | 7 | 3impib 1115 | 1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ifcif 4531 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-if 4532 |
This theorem is referenced by: dedth3v 4594 faclbnd4lem2 14330 dvdsle 16344 gcdaddm 16559 ipdiri 30859 hvaddcan 31099 hvsubadd 31106 norm3dif 31179 omlsii 31432 chjass 31562 ledi 31569 spansncv 31682 pjcjt2 31721 pjopyth 31749 hoaddass 31811 hocsubdir 31814 hoddi 32019 |
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