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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 4518. (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 4518 | . . 3 ⊢ ((𝜒 ∧ 𝜃) → 𝜁) |
9 | 2, 4, 8 | dedth2h 4518 | . 2 ⊢ ((𝜑 ∧ 𝜓) → ((𝜒 ∧ 𝜃) → 𝜏)) |
10 | 9 | imp 407 | 1 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) → 𝜏) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ifcif 4459 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-if 4460 |
This theorem is referenced by: dedth4v 4523 fprg 7027 omopth 8492 nn0opth2 13986 ax5seglem8 27304 hvsubsub4 29422 norm3lemt 29514 eigorth 30200 |
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