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Theorem fvifdc 5270
Description: Move a conditional outside of a function. (Contributed by Jim Kingdon, 1-Jan-2022.)
Assertion
Ref Expression
fvifdc (DECID 𝜑 → (𝐹‘if(𝜑, 𝐴, 𝐵)) = if(𝜑, (𝐹𝐴), (𝐹𝐵)))

Proof of Theorem fvifdc
StepHypRef Expression
1 fveq2 5251 . 2 (if(𝜑, 𝐴, 𝐵) = 𝐴 → (𝐹‘if(𝜑, 𝐴, 𝐵)) = (𝐹𝐴))
2 fveq2 5251 . 2 (if(𝜑, 𝐴, 𝐵) = 𝐵 → (𝐹‘if(𝜑, 𝐴, 𝐵)) = (𝐹𝐵))
31, 2ifsbdc 3385 1 (DECID 𝜑 → (𝐹‘if(𝜑, 𝐴, 𝐵)) = if(𝜑, (𝐹𝐴), (𝐹𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  DECID wdc 776   = wceq 1285  ifcif 3373  cfv 4967
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-rex 2359  df-v 2614  df-un 2988  df-if 3374  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-br 3812  df-iota 4932  df-fv 4975
This theorem is referenced by: (None)
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