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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 7428 | . 2 ⊢ (if(𝜑, 𝐵, 𝐶) = 𝐵 → (𝐴𝐹if(𝜑, 𝐵, 𝐶)) = (𝐴𝐹𝐵)) | |
2 | oveq2 7428 | . 2 ⊢ (if(𝜑, 𝐵, 𝐶) = 𝐶 → (𝐴𝐹if(𝜑, 𝐵, 𝐶)) = (𝐴𝐹𝐶)) | |
3 | 1, 2 | ifsb 4542 | 1 ⊢ (𝐴𝐹if(𝜑, 𝐵, 𝐶)) = if(𝜑, (𝐴𝐹𝐵), (𝐴𝐹𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ifcif 4529 (class class class)co 7420 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-iota 6500 df-fv 6556 df-ov 7423 |
This theorem is referenced by: ramcl 16997 matsc 22351 scmatscmide 22408 mulmarep1el 22473 maducoeval2 22541 madugsum 22544 itg2const 25669 itg2monolem1 25679 iblmulc2 25759 itgmulc2lem1 25760 bddmulibl 25767 dchrvmasumiflem2 27434 rpvmasum2 27444 sgnneg 34160 itg2addnclem 37144 itgaddnclem2 37152 itgmulc2nclem1 37159 selvvvval 41818 sqrtcval2 43072 |
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