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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 7166 | . 2 ⊢ (if(𝜑, 𝐵, 𝐶) = 𝐵 → (𝐴𝐹if(𝜑, 𝐵, 𝐶)) = (𝐴𝐹𝐵)) | |
2 | oveq2 7166 | . 2 ⊢ (if(𝜑, 𝐵, 𝐶) = 𝐶 → (𝐴𝐹if(𝜑, 𝐵, 𝐶)) = (𝐴𝐹𝐶)) | |
3 | 1, 2 | ifsb 4482 | 1 ⊢ (𝐴𝐹if(𝜑, 𝐵, 𝐶)) = if(𝜑, (𝐴𝐹𝐵), (𝐴𝐹𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ifcif 4469 (class class class)co 7158 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-iota 6316 df-fv 6365 df-ov 7161 |
This theorem is referenced by: ramcl 16367 matsc 21061 scmatscmide 21118 mulmarep1el 21183 maducoeval2 21251 madugsum 21254 itg2const 24343 itg2monolem1 24353 iblmulc2 24433 itgmulc2lem1 24434 bddmulibl 24441 dchrvmasumiflem2 26080 rpvmasum2 26090 sgnneg 31800 itg2addnclem 34945 itgaddnclem2 34953 itgmulc2nclem1 34960 |
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