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Mirrors > Home > MPE Home > Th. List > dedth4v | Structured version Visualization version GIF version |
Description: Weak deduction theorem for eliminating a hypothesis with 4 class variables. See comments in dedth2v 4552. (Contributed by NM, 21-Apr-2007.) (Proof shortened by Eric Schmidt, 28-Jul-2009.) |
Ref | Expression |
---|---|
dedth4v.1 | ⊢ (𝐴 = if(𝜑, 𝐴, 𝑅) → (𝜓 ↔ 𝜒)) |
dedth4v.2 | ⊢ (𝐵 = if(𝜑, 𝐵, 𝑆) → (𝜒 ↔ 𝜃)) |
dedth4v.3 | ⊢ (𝐶 = if(𝜑, 𝐶, 𝑇) → (𝜃 ↔ 𝜏)) |
dedth4v.4 | ⊢ (𝐷 = if(𝜑, 𝐷, 𝑈) → (𝜏 ↔ 𝜂)) |
dedth4v.5 | ⊢ 𝜂 |
Ref | Expression |
---|---|
dedth4v | ⊢ (𝜑 → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dedth4v.1 | . . . 4 ⊢ (𝐴 = if(𝜑, 𝐴, 𝑅) → (𝜓 ↔ 𝜒)) | |
2 | dedth4v.2 | . . . 4 ⊢ (𝐵 = if(𝜑, 𝐵, 𝑆) → (𝜒 ↔ 𝜃)) | |
3 | dedth4v.3 | . . . 4 ⊢ (𝐶 = if(𝜑, 𝐶, 𝑇) → (𝜃 ↔ 𝜏)) | |
4 | dedth4v.4 | . . . 4 ⊢ (𝐷 = if(𝜑, 𝐷, 𝑈) → (𝜏 ↔ 𝜂)) | |
5 | dedth4v.5 | . . . 4 ⊢ 𝜂 | |
6 | 1, 2, 3, 4, 5 | dedth4h 4551 | . . 3 ⊢ (((𝜑 ∧ 𝜑) ∧ (𝜑 ∧ 𝜑)) → 𝜓) |
7 | 6 | anidms 568 | . 2 ⊢ ((𝜑 ∧ 𝜑) → 𝜓) |
8 | 7 | anidms 568 | 1 ⊢ (𝜑 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ifcif 4490 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-if 4491 |
This theorem is referenced by: (None) |
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