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Theorem ifsbdc 3492
 Description: Distribute a function over an if-clause. (Contributed by Jim Kingdon, 1-Jan-2022.)
Hypotheses
Ref Expression
ifsbdc.1
ifsbdc.2
Assertion
Ref Expression
ifsbdc DECID

Proof of Theorem ifsbdc
StepHypRef Expression
1 exmiddc 822 . 2 DECID
2 iftrue 3485 . . . . 5
3 ifsbdc.1 . . . . 5
42, 3syl 14 . . . 4
5 iftrue 3485 . . . 4
64, 5eqtr4d 2176 . . 3
7 iffalse 3488 . . . . 5
8 ifsbdc.2 . . . . 5
97, 8syl 14 . . . 4
10 iffalse 3488 . . . 4
119, 10eqtr4d 2176 . . 3
126, 11jaoi 706 . 2
131, 12syl 14 1 DECID
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wo 698  DECID wdc 820   wceq 1332  cif 3480 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-dc 821  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-if 3481 This theorem is referenced by:  fvifdc  5452
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