Theorem ovif 7525

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 7433	. 2 ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐴 → (if(𝜑, 𝐴, 𝐵)𝐹𝐶) = (𝐴𝐹𝐶))
2		oveq1 7433	. 2 ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐵 → (if(𝜑, 𝐴, 𝐵)𝐹𝐶) = (𝐵𝐹𝐶))
3	1, 2	ifsb 4546	1 ⊢ (if(𝜑, 𝐴, 𝐵)𝐹𝐶) = if(𝜑, (𝐴𝐹𝐶), (𝐵𝐹𝐶))

Syntax hints:   = wceq 1534  ifcif 4533  (class class class)co 7426

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4916  df-br 5156  df-iota 6508  df-fv 6564  df-ov 7429

This theorem is referenced by:  scmatscm  22509  pmatcollpwscmatlem1  22785  idpm2idmp  22797  monmat2matmon  22820  chmatval  22825  leibpi  26973  musumsum  27223  muinv  27224  dchrinvcl  27285  rpvmasum2  27544  padicabvcxp  27664  pnfneige0  33768  plymulx0  34395  ftc1anclem6  37401  reabssgn  43321  sqrtcval  43326  linc0scn0  47824