Theorem ovif 7451

Description: Move a conditional outside of an operation. (Contributed by Thierry Arnoux, 25-Jan-2017.)

Assertion

Ref	Expression
ovif	⊢ (if(𝜑, 𝐴, 𝐵)𝐹𝐶) = if(𝜑, (𝐴𝐹𝐶), (𝐵𝐹𝐶))

Proof of Theorem ovif

Step	Hyp	Ref	Expression
1		oveq1 7360	. 2 ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐴 → (if(𝜑, 𝐴, 𝐵)𝐹𝐶) = (𝐴𝐹𝐶))
2		oveq1 7360	. 2 ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐵 → (if(𝜑, 𝐴, 𝐵)𝐹𝐶) = (𝐵𝐹𝐶))
3	1, 2	ifsb 4492	1 ⊢ (if(𝜑, 𝐴, 𝐵)𝐹𝐶) = if(𝜑, (𝐴𝐹𝐶), (𝐵𝐹𝐶))

Syntax hints:   = wceq 1540  ifcif 4478  (class class class)co 7353

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-iota 6442  df-fv 6494  df-ov 7356

This theorem is referenced by:  scmatscm  22417  pmatcollpwscmatlem1  22693  idpm2idmp  22705  monmat2matmon  22728  chmatval  22733  leibpi  26869  musumsum  27119  muinv  27120  dchrinvcl  27181  rpvmasum2  27440  padicabvcxp  27560  pnfneige0  33937  plymulx0  34534  ftc1anclem6  37697  reabssgn  43629  sqrtcval  43634  linc0scn0  48428