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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 7366 | . 2 ⊢ (if(𝜑, 𝐵, 𝐶) = 𝐵 → (𝐴𝐹if(𝜑, 𝐵, 𝐶)) = (𝐴𝐹𝐵)) | |
2 | oveq2 7366 | . 2 ⊢ (if(𝜑, 𝐵, 𝐶) = 𝐶 → (𝐴𝐹if(𝜑, 𝐵, 𝐶)) = (𝐴𝐹𝐶)) | |
3 | 1, 2 | ifsb 4500 | 1 ⊢ (𝐴𝐹if(𝜑, 𝐵, 𝐶)) = if(𝜑, (𝐴𝐹𝐵), (𝐴𝐹𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ifcif 4487 (class class class)co 7358 |
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-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-iota 6449 df-fv 6505 df-ov 7361 |
This theorem is referenced by: ramcl 16906 matsc 21815 scmatscmide 21872 mulmarep1el 21937 maducoeval2 22005 madugsum 22008 itg2const 25121 itg2monolem1 25131 iblmulc2 25211 itgmulc2lem1 25212 bddmulibl 25219 dchrvmasumiflem2 26866 rpvmasum2 26876 sgnneg 33197 itg2addnclem 36175 itgaddnclem2 36183 itgmulc2nclem1 36190 mhphf 40814 sqrtcval2 42002 |
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