Theorem ovif 7489

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 7398	. 2 ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐴 → (if(𝜑, 𝐴, 𝐵)𝐹𝐶) = (𝐴𝐹𝐶))
2		oveq1 7398	. 2 ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐵 → (if(𝜑, 𝐴, 𝐵)𝐹𝐶) = (𝐵𝐹𝐶))
3	1, 2	ifsb 4491	1 ⊢ (if(𝜑, 𝐴, 𝐵)𝐹𝐶) = if(𝜑, (𝐴𝐹𝐶), (𝐵𝐹𝐶))

Syntax hints:   = wceq 1559  ifcif 4477  (class class class)co 7391

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-iota 6472  df-fv 6524  df-ov 7394

This theorem is referenced by:  scmatscm  22561  pmatcollpwscmatlem1  22837  idpm2idmp  22849  monmat2matmon  22872  chmatval  22877  plyn0mulidp  26333  leibpi  26995  musumsum  27244  muinv  27245  dchrinvcl  27305  rpvmasum2  27564  padicabvcxp  27684  mplmulmvr  33797  pnfneige0  34209  ftc1anclem6  38158  reabssgn  44173  sqrtcval  44178  linc0scn0  49006