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Theorem ovif2 7242
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 7154 . 2 (if(𝜑, 𝐵, 𝐶) = 𝐵 → (𝐴𝐹if(𝜑, 𝐵, 𝐶)) = (𝐴𝐹𝐵))
2 oveq2 7154 . 2 (if(𝜑, 𝐵, 𝐶) = 𝐶 → (𝐴𝐹if(𝜑, 𝐵, 𝐶)) = (𝐴𝐹𝐶))
31, 2ifsb 4463 1 (𝐴𝐹if(𝜑, 𝐵, 𝐶)) = if(𝜑, (𝐴𝐹𝐵), (𝐴𝐹𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  ifcif 4450  (class class class)co 7146
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-v 3482  df-un 3924  df-in 3926  df-ss 3936  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-br 5054  df-iota 6303  df-fv 6352  df-ov 7149
This theorem is referenced by:  ramcl  16361  matsc  21054  scmatscmide  21111  mulmarep1el  21176  maducoeval2  21244  madugsum  21247  itg2const  24342  itg2monolem1  24352  iblmulc2  24432  itgmulc2lem1  24433  bddmulibl  24440  dchrvmasumiflem2  26084  rpvmasum2  26094  sgnneg  31825  itg2addnclem  35020  itgaddnclem2  35028  itgmulc2nclem1  35035  sqrtcval2  40198
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