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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 4124 | . 2 ⊢ ((𝜓 ∧ 𝜒) → 𝜃) |
6 | 1, 2, 5 | mp2an 708 | 1 ⊢ 𝜃 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 = wceq 1483 ifcif 4086 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-if 4087 |
This theorem is referenced by: keepel 4155 boxcutc 7951 fin23lem13 9154 abvtrivd 18840 znf1o 19900 zntoslem 19905 dscmet 22377 sqff1o 24908 lgsne0 25060 dchrisum0flblem1 25197 dchrisum0flblem2 25198 |
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