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Mirrors > Home > MPE Home > Th. List > dedth2v | Structured version Visualization version GIF version |
Description: Weak deduction theorem for eliminating a hypothesis with 2 class variables. Note: if the hypothesis can be separated into two hypotheses, each with one class variable, then dedth2h 4549 is simpler to use. See also comments in dedth 4548. (Contributed by NM, 13-Aug-1999.) (Proof shortened by Eric Schmidt, 28-Jul-2009.) |
Ref | Expression |
---|---|
dedth2v.1 | ⊢ (𝐴 = if(𝜑, 𝐴, 𝐶) → (𝜓 ↔ 𝜒)) |
dedth2v.2 | ⊢ (𝐵 = if(𝜑, 𝐵, 𝐷) → (𝜒 ↔ 𝜃)) |
dedth2v.3 | ⊢ 𝜃 |
Ref | Expression |
---|---|
dedth2v | ⊢ (𝜑 → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dedth2v.1 | . . 3 ⊢ (𝐴 = if(𝜑, 𝐴, 𝐶) → (𝜓 ↔ 𝜒)) | |
2 | dedth2v.2 | . . 3 ⊢ (𝐵 = if(𝜑, 𝐵, 𝐷) → (𝜒 ↔ 𝜃)) | |
3 | dedth2v.3 | . . 3 ⊢ 𝜃 | |
4 | 1, 2, 3 | dedth2h 4549 | . 2 ⊢ ((𝜑 ∧ 𝜑) → 𝜓) |
5 | 4 | anidms 568 | 1 ⊢ (𝜑 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = 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: ltweuz 13875 omlsi 30395 pjhfo 30697 |
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