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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 4074 | . 2 ⊢ ((𝜓 ∧ 𝜒) → 𝜃) |
6 | 1, 2, 5 | mp2an 704 | 1 ⊢ 𝜃 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ifcif 4036 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-if 4037 |
This theorem is referenced by: keepel 4105 boxcutc 7837 fin23lem13 9037 abvtrivd 18663 znf1o 19719 zntoslem 19724 dscmet 22187 sqff1o 24708 lgsne0 24860 dchrisum0flblem1 24997 dchrisum0flblem2 24998 |
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