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Mirrors > Home > MPE Home > Th. List > dedth3v | Structured version Visualization version GIF version |
Description: Weak deduction theorem for eliminating a hypothesis with 3 class variables. See comments in dedth2v 4530. (Contributed by NM, 13-Aug-1999.) (Proof shortened by Eric Schmidt, 28-Jul-2009.) |
Ref | Expression |
---|---|
dedth3v.1 | ⊢ (𝐴 = if(𝜑, 𝐴, 𝐷) → (𝜓 ↔ 𝜒)) |
dedth3v.2 | ⊢ (𝐵 = if(𝜑, 𝐵, 𝑅) → (𝜒 ↔ 𝜃)) |
dedth3v.3 | ⊢ (𝐶 = if(𝜑, 𝐶, 𝑆) → (𝜃 ↔ 𝜏)) |
dedth3v.4 | ⊢ 𝜏 |
Ref | Expression |
---|---|
dedth3v | ⊢ (𝜑 → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dedth3v.1 | . . . 4 ⊢ (𝐴 = if(𝜑, 𝐴, 𝐷) → (𝜓 ↔ 𝜒)) | |
2 | dedth3v.2 | . . . 4 ⊢ (𝐵 = if(𝜑, 𝐵, 𝑅) → (𝜒 ↔ 𝜃)) | |
3 | dedth3v.3 | . . . 4 ⊢ (𝐶 = if(𝜑, 𝐶, 𝑆) → (𝜃 ↔ 𝜏)) | |
4 | dedth3v.4 | . . . 4 ⊢ 𝜏 | |
5 | 1, 2, 3, 4 | dedth3h 4528 | . . 3 ⊢ ((𝜑 ∧ 𝜑 ∧ 𝜑) → 𝜓) |
6 | 5 | 3anidm12 1415 | . 2 ⊢ ((𝜑 ∧ 𝜑) → 𝜓) |
7 | 6 | anidms 569 | 1 ⊢ (𝜑 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1536 ifcif 4470 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-ex 1780 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-if 4471 |
This theorem is referenced by: sseliALT 5216 |
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