Theorem ovif2 7460

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 7369	. 2 ⊢ (if(𝜑, 𝐵, 𝐶) = 𝐵 → (𝐴𝐹if(𝜑, 𝐵, 𝐶)) = (𝐴𝐹𝐵))
2		oveq2 7369	. 2 ⊢ (if(𝜑, 𝐵, 𝐶) = 𝐶 → (𝐴𝐹if(𝜑, 𝐵, 𝐶)) = (𝐴𝐹𝐶))
3	1, 2	ifsb 4481	1 ⊢ (𝐴𝐹if(𝜑, 𝐵, 𝐶)) = if(𝜑, (𝐴𝐹𝐵), (𝐴𝐹𝐶))

Syntax hints:   = wceq 1542  ifcif 4467  (class class class)co 7361

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-iota 6449  df-fv 6501  df-ov 7364

This theorem is referenced by:  ramcl  16994  psrascl  21970  psdmvr  22148  matsc  22428  scmatscmide  22485  mulmarep1el  22550  maducoeval2  22618  madugsum  22621  itg2const  25720  itg2monolem1  25730  iblmulc2  25811  itgmulc2lem1  25812  bddmulibl  25819  dchrvmasumiflem2  27482  rpvmasum2  27492  sgnneg  32924  itg2addnclem  38009  itgaddnclem2  38017  itgmulc2nclem1  38024  readvrec  42811  selvvvval  43035  sqrtcval2  44090