Theorem ovif 7487

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 7394	. 2 ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐴 → (if(𝜑, 𝐴, 𝐵)𝐹𝐶) = (𝐴𝐹𝐶))
2		oveq1 7394	. 2 ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐵 → (if(𝜑, 𝐴, 𝐵)𝐹𝐶) = (𝐵𝐹𝐶))
3	1, 2	ifsb 4502	1 ⊢ (if(𝜑, 𝐴, 𝐵)𝐹𝐶) = if(𝜑, (𝐴𝐹𝐶), (𝐵𝐹𝐶))

Syntax hints:   = wceq 1540  ifcif 4488  (class class class)co 7387

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 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-iota 6464  df-fv 6519  df-ov 7390

This theorem is referenced by:  scmatscm  22400  pmatcollpwscmatlem1  22676  idpm2idmp  22688  monmat2matmon  22711  chmatval  22716  leibpi  26852  musumsum  27102  muinv  27103  dchrinvcl  27164  rpvmasum2  27423  padicabvcxp  27543  pnfneige0  33941  plymulx0  34538  ftc1anclem6  37692  reabssgn  43625  sqrtcval  43630  linc0scn0  48412