Theorem ovif2 7373

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

Step	Hyp	Ref	Expression
1		oveq2 7283	. 2 ⊢ (if(𝜑, 𝐵, 𝐶) = 𝐵 → (𝐴𝐹if(𝜑, 𝐵, 𝐶)) = (𝐴𝐹𝐵))
2		oveq2 7283	. 2 ⊢ (if(𝜑, 𝐵, 𝐶) = 𝐶 → (𝐴𝐹if(𝜑, 𝐵, 𝐶)) = (𝐴𝐹𝐶))
3	1, 2	ifsb 4472	1 ⊢ (𝐴𝐹if(𝜑, 𝐵, 𝐶)) = if(𝜑, (𝐴𝐹𝐵), (𝐴𝐹𝐶))

Syntax hints:   = wceq 1539  ifcif 4459  (class class class)co 7275

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-iota 6391  df-fv 6441  df-ov 7278

This theorem is referenced by:  ramcl  16730  matsc  21599  scmatscmide  21656  mulmarep1el  21721  maducoeval2  21789  madugsum  21792  itg2const  24905  itg2monolem1  24915  iblmulc2  24995  itgmulc2lem1  24996  bddmulibl  25003  dchrvmasumiflem2  26650  rpvmasum2  26660  sgnneg  32507  itg2addnclem  35828  itgaddnclem2  35836  itgmulc2nclem1  35843  mhphf  40285  sqrtcval2  41250