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Mirrors > Home > MPE Home > Th. List > keephyp | Structured version Visualization version GIF version |
Description: Transform a hypothesis 𝜓 that we want to keep (but contains the same class variable 𝐴 used in the eliminated hypothesis) for use with the weak deduction theorem. (Contributed by NM, 15-May-1999.) |
Ref | Expression |
---|---|
keephyp.1 | ⊢ (𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜓 ↔ 𝜃)) |
keephyp.2 | ⊢ (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒 ↔ 𝜃)) |
keephyp.3 | ⊢ 𝜓 |
keephyp.4 | ⊢ 𝜒 |
Ref | Expression |
---|---|
keephyp | ⊢ 𝜃 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | keephyp.3 | . 2 ⊢ 𝜓 | |
2 | keephyp.4 | . 2 ⊢ 𝜒 | |
3 | keephyp.1 | . . 3 ⊢ (𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜓 ↔ 𝜃)) | |
4 | keephyp.2 | . . 3 ⊢ (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒 ↔ 𝜃)) | |
5 | 3, 4 | ifboth 4529 | . 2 ⊢ ((𝜓 ∧ 𝜒) → 𝜃) |
6 | 1, 2, 5 | mp2an 691 | 1 ⊢ 𝜃 |
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: boxcutc 8885 fin23lem13 10276 abvtrivd 20342 znf1o 20981 zntoslem 20986 dscmet 23951 sqff1o 26554 lgsne0 26706 dchrisum0flblem1 26879 dchrisum0flblem2 26880 |
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