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Theorem ovif2 7454
Description: Move a conditional outside of an operation. (Contributed by Thierry Arnoux, 1-Oct-2018.)
Assertion
Ref Expression
ovif2 (𝐴𝐹if(𝜑, 𝐵, 𝐶)) = if(𝜑, (𝐴𝐹𝐵), (𝐴𝐹𝐶))

Proof of Theorem ovif2
StepHypRef Expression
1 oveq2 7364 . 2 (if(𝜑, 𝐵, 𝐶) = 𝐵 → (𝐴𝐹if(𝜑, 𝐵, 𝐶)) = (𝐴𝐹𝐵))
2 oveq2 7364 . 2 (if(𝜑, 𝐵, 𝐶) = 𝐶 → (𝐴𝐹if(𝜑, 𝐵, 𝐶)) = (𝐴𝐹𝐶))
31, 2ifsb 4499 1 (𝐴𝐹if(𝜑, 𝐵, 𝐶)) = if(𝜑, (𝐴𝐹𝐵), (𝐴𝐹𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541  ifcif 4486  (class class class)co 7356
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-iota 6448  df-fv 6504  df-ov 7359
This theorem is referenced by:  ramcl  16900  matsc  21797  scmatscmide  21854  mulmarep1el  21919  maducoeval2  21987  madugsum  21990  itg2const  25103  itg2monolem1  25113  iblmulc2  25193  itgmulc2lem1  25194  bddmulibl  25201  dchrvmasumiflem2  26848  rpvmasum2  26858  sgnneg  33080  itg2addnclem  36119  itgaddnclem2  36127  itgmulc2nclem1  36134  mhphf  40748  sqrtcval2  41895
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