Theorem ovif2 7549

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 7456	. 2 ⊢ (if(𝜑, 𝐵, 𝐶) = 𝐵 → (𝐴𝐹if(𝜑, 𝐵, 𝐶)) = (𝐴𝐹𝐵))
2		oveq2 7456	. 2 ⊢ (if(𝜑, 𝐵, 𝐶) = 𝐶 → (𝐴𝐹if(𝜑, 𝐵, 𝐶)) = (𝐴𝐹𝐶))
3	1, 2	ifsb 4561	1 ⊢ (𝐴𝐹if(𝜑, 𝐵, 𝐶)) = if(𝜑, (𝐴𝐹𝐵), (𝐴𝐹𝐶))

Syntax hints:   = wceq 1537  ifcif 4548  (class class class)co 7448

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-iota 6525  df-fv 6581  df-ov 7451

This theorem is referenced by:  ramcl  17076  psrascl  22022  matsc  22477  scmatscmide  22534  mulmarep1el  22599  maducoeval2  22667  madugsum  22670  itg2const  25795  itg2monolem1  25805  iblmulc2  25886  itgmulc2lem1  25887  bddmulibl  25894  dchrvmasumiflem2  27564  rpvmasum2  27574  sgnneg  34505  itg2addnclem  37631  itgaddnclem2  37639  itgmulc2nclem1  37646  selvvvval  42540  sqrtcval2  43604