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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 4464 | . 2 ⊢ ((𝜓 ∧ 𝜒) → 𝜃) |
6 | 1, 2, 5 | mp2an 692 | 1 ⊢ 𝜃 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1543 ifcif 4425 |
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 2018 ax-8 2114 ax-9 2122 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-if 4426 |
This theorem is referenced by: boxcutc 8600 fin23lem13 9911 abvtrivd 19830 znf1o 20470 zntoslem 20475 dscmet 23424 sqff1o 26018 lgsne0 26170 dchrisum0flblem1 26343 dchrisum0flblem2 26344 |
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