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Mirrors > Home > MPE Home > Th. List > dedth4h | Structured version Visualization version GIF version |
Description: Weak deduction theorem eliminating four hypotheses. See comments in dedth2h 4549. (Contributed by NM, 16-May-1999.) |
Ref | Expression |
---|---|
dedth4h.1 | ⊢ (𝐴 = if(𝜑, 𝐴, 𝑅) → (𝜏 ↔ 𝜂)) |
dedth4h.2 | ⊢ (𝐵 = if(𝜓, 𝐵, 𝑆) → (𝜂 ↔ 𝜁)) |
dedth4h.3 | ⊢ (𝐶 = if(𝜒, 𝐶, 𝐹) → (𝜁 ↔ 𝜎)) |
dedth4h.4 | ⊢ (𝐷 = if(𝜃, 𝐷, 𝐺) → (𝜎 ↔ 𝜌)) |
dedth4h.5 | ⊢ 𝜌 |
Ref | Expression |
---|---|
dedth4h | ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) → 𝜏) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dedth4h.1 | . . . 4 ⊢ (𝐴 = if(𝜑, 𝐴, 𝑅) → (𝜏 ↔ 𝜂)) | |
2 | 1 | imbi2d 341 | . . 3 ⊢ (𝐴 = if(𝜑, 𝐴, 𝑅) → (((𝜒 ∧ 𝜃) → 𝜏) ↔ ((𝜒 ∧ 𝜃) → 𝜂))) |
3 | dedth4h.2 | . . . 4 ⊢ (𝐵 = if(𝜓, 𝐵, 𝑆) → (𝜂 ↔ 𝜁)) | |
4 | 3 | imbi2d 341 | . . 3 ⊢ (𝐵 = if(𝜓, 𝐵, 𝑆) → (((𝜒 ∧ 𝜃) → 𝜂) ↔ ((𝜒 ∧ 𝜃) → 𝜁))) |
5 | dedth4h.3 | . . . 4 ⊢ (𝐶 = if(𝜒, 𝐶, 𝐹) → (𝜁 ↔ 𝜎)) | |
6 | dedth4h.4 | . . . 4 ⊢ (𝐷 = if(𝜃, 𝐷, 𝐺) → (𝜎 ↔ 𝜌)) | |
7 | dedth4h.5 | . . . 4 ⊢ 𝜌 | |
8 | 5, 6, 7 | dedth2h 4549 | . . 3 ⊢ ((𝜒 ∧ 𝜃) → 𝜁) |
9 | 2, 4, 8 | dedth2h 4549 | . 2 ⊢ ((𝜑 ∧ 𝜓) → ((𝜒 ∧ 𝜃) → 𝜏)) |
10 | 9 | imp 408 | 1 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) → 𝜏) |
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: dedth4v 4554 fprg 7105 omopth 8612 nn0opth2 14181 ax5seglem8 27934 hvsubsub4 30051 norm3lemt 30143 eigorth 30829 |
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