Theorem ovif2 7510

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 7419	. 2 ⊢ (if(𝜑, 𝐵, 𝐶) = 𝐵 → (𝐴𝐹if(𝜑, 𝐵, 𝐶)) = (𝐴𝐹𝐵))
2		oveq2 7419	. 2 ⊢ (if(𝜑, 𝐵, 𝐶) = 𝐶 → (𝐴𝐹if(𝜑, 𝐵, 𝐶)) = (𝐴𝐹𝐶))
3	1, 2	ifsb 4506	1 ⊢ (𝐴𝐹if(𝜑, 𝐵, 𝐶)) = if(𝜑, (𝐴𝐹𝐵), (𝐴𝐹𝐶))

Syntax hints:   = wceq 1567  ifcif 4492  (class class class)co 7411

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-iota 6493  df-fv 6545  df-ov 7414

This theorem is referenced by:  sgnneg  15136  ramcl  17088  psrascl  22096  selvvvval  22261  psdmvr  22300  matsc  22575  scmatscmide  22632  mulmarep1el  22697  maducoeval2  22765  madugsum  22768  itg2const  25867  itg2monolem1  25877  iblmulc2  25958  itgmulc2lem1  25959  bddmulibl  25966  dchrvmasumiflem2  27631  rpvmasum2  27641  itg2addnclem  38209  itgaddnclem2  38217  itgmulc2nclem1  38224  readvrec  43012  sqrtcval2  44259