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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 4506. (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 4504 | . . 3 ⊢ ((𝜑 ∧ 𝜑 ∧ 𝜑) → 𝜓) |
6 | 5 | 3anidm12 1421 | . 2 ⊢ ((𝜑 ∧ 𝜑) → 𝜓) |
7 | 6 | anidms 570 | 1 ⊢ (𝜑 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1543 ifcif 4444 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-if 4445 |
This theorem is referenced by: sseliALT 5207 |
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