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Theorem fvifdc 5249
Description: Move a conditional outside of a function. (Contributed by Jim Kingdon, 1-Jan-2022.)
Assertion
Ref Expression
fvifdc  |-  (DECID  ph  ->  ( F `  if (
ph ,  A ,  B ) )  =  if ( ph , 
( F `  A
) ,  ( F `
 B ) ) )

Proof of Theorem fvifdc
StepHypRef Expression
1 fveq2 5230 . 2  |-  ( if ( ph ,  A ,  B )  =  A  ->  ( F `  if ( ph ,  A ,  B ) )  =  ( F `  A
) )
2 fveq2 5230 . 2  |-  ( if ( ph ,  A ,  B )  =  B  ->  ( F `  if ( ph ,  A ,  B ) )  =  ( F `  B
) )
31, 2ifsbdc 3380 1  |-  (DECID  ph  ->  ( F `  if (
ph ,  A ,  B ) )  =  if ( ph , 
( F `  A
) ,  ( F `
 B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4  DECID wdc 776    = wceq 1285   ifcif 3368   ` cfv 4952
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 2612  df-un 2986  df-if 3369  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-iota 4917  df-fv 4960
This theorem is referenced by: (None)
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