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Mirrors > Home > MPE Home > Th. List > ovif2 | Structured version Visualization version GIF version |
Description: Move a conditional outside of an operation. (Contributed by Thierry Arnoux, 1-Oct-2018.) |
Ref | Expression |
---|---|
ovif2 | ⊢ (𝐴𝐹if(𝜑, 𝐵, 𝐶)) = if(𝜑, (𝐴𝐹𝐵), (𝐴𝐹𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 7439 | . 2 ⊢ (if(𝜑, 𝐵, 𝐶) = 𝐵 → (𝐴𝐹if(𝜑, 𝐵, 𝐶)) = (𝐴𝐹𝐵)) | |
2 | oveq2 7439 | . 2 ⊢ (if(𝜑, 𝐵, 𝐶) = 𝐶 → (𝐴𝐹if(𝜑, 𝐵, 𝐶)) = (𝐴𝐹𝐶)) | |
3 | 1, 2 | ifsb 4544 | 1 ⊢ (𝐴𝐹if(𝜑, 𝐵, 𝐶)) = if(𝜑, (𝐴𝐹𝐵), (𝐴𝐹𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ifcif 4531 (class class class)co 7431 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-iota 6516 df-fv 6571 df-ov 7434 |
This theorem is referenced by: ramcl 17063 psrascl 22017 matsc 22472 scmatscmide 22529 mulmarep1el 22594 maducoeval2 22662 madugsum 22665 itg2const 25790 itg2monolem1 25800 iblmulc2 25881 itgmulc2lem1 25882 bddmulibl 25889 dchrvmasumiflem2 27561 rpvmasum2 27571 sgnneg 34522 itg2addnclem 37658 itgaddnclem2 37666 itgmulc2nclem1 37673 readvrec 42371 selvvvval 42572 sqrtcval2 43632 |
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