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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 4520 is simpler to use. See also comments in dedth 4519. (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 4520 | . 2 ⊢ ((𝜑 ∧ 𝜑) → 𝜓) |
5 | 4 | anidms 567 | 1 ⊢ (𝜑 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 = wceq 1528 ifcif 4463 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-ex 1772 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-if 4464 |
This theorem is referenced by: ltweuz 13317 omlsi 29108 pjhfo 29410 |
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