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Theorem fvif 6659
Description: Move a conditional outside of a function. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
fvif (𝐹‘if(𝜑, 𝐴, 𝐵)) = if(𝜑, (𝐹𝐴), (𝐹𝐵))

Proof of Theorem fvif
StepHypRef Expression
1 fveq2 6643 . 2 (if(𝜑, 𝐴, 𝐵) = 𝐴 → (𝐹‘if(𝜑, 𝐴, 𝐵)) = (𝐹𝐴))
2 fveq2 6643 . 2 (if(𝜑, 𝐴, 𝐵) = 𝐵 → (𝐹‘if(𝜑, 𝐴, 𝐵)) = (𝐹𝐵))
31, 2ifsb 4453 1 (𝐹‘if(𝜑, 𝐴, 𝐵)) = if(𝜑, (𝐹𝐴), (𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  ifcif 4440  cfv 6328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-rab 3135  df-v 3473  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-iota 6287  df-fv 6336
This theorem is referenced by:  ccatco  14176  sumeq2ii  15029  prodeq2ii  15246  ruclem1  15563  xpsrnbas  16822  mat2pmat1  21315  decpmatid  21353  pmatcollpwscmatlem1  21372  copco  23601  pcopt  23605  pcopt2  23606  limccnp  24472  prmorcht  25741  pclogsum  25777  mblfinlem2  34975  ftc1anclem8  35017  ftc1anc  35018  fvifeq  43629
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