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Theorem ovif2 6780
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 6698 . 2 (if(𝜑, 𝐵, 𝐶) = 𝐵 → (𝐴𝐹if(𝜑, 𝐵, 𝐶)) = (𝐴𝐹𝐵))
2 oveq2 6698 . 2 (if(𝜑, 𝐵, 𝐶) = 𝐶 → (𝐴𝐹if(𝜑, 𝐵, 𝐶)) = (𝐴𝐹𝐶))
31, 2ifsb 4132 1 (𝐴𝐹if(𝜑, 𝐵, 𝐶)) = if(𝜑, (𝐴𝐹𝐵), (𝐴𝐹𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1523  ifcif 4119  (class class class)co 6690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-iota 5889  df-fv 5934  df-ov 6693
This theorem is referenced by:  ramcl  15780  matsc  20304  scmatscmide  20361  mulmarep1el  20426  maducoeval2  20494  madugsum  20497  itg2const  23552  itg2monolem1  23562  iblmulc2  23642  itgmulc2lem1  23643  bddmulibl  23650  dchrvmasumiflem2  25236  rpvmasum2  25246  sgnneg  30730  itg2addnclem  33591  itgaddnclem2  33599  itgmulc2nclem1  33606
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