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Theorem decidin 12993
 Description: If A is a decidable subclass of B (meaning: it is a subclass of B and it is decidable in B), and B is decidable in C, then A is decidable in C. (Contributed by BJ, 19-Feb-2022.)
Hypotheses
Ref Expression
decidin.ss
decidin.a DECIDin
decidin.b DECIDin
Assertion
Ref Expression
decidin DECIDin

Proof of Theorem decidin
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 decidin.b . . . 4 DECIDin
2 decidi 12991 . . . 4 DECIDin
31, 2syl 14 . . 3
4 decidin.a . . . . 5 DECIDin
5 decidi 12991 . . . . 5 DECIDin
64, 5syl 14 . . . 4
7 decidin.ss . . . . . 6
87ssneld 3094 . . . . 5
9 olc 700 . . . . 5
108, 9syl6 33 . . . 4
116, 10jaod 706 . . 3
123, 11syld 45 . 2
1312decidr 12992 1 DECIDin
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wo 697   wcel 1480   wss 3066   DECIDin wdcin 12989 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-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119 This theorem depends on definitions:  df-bi 116  df-dc 820  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-v 2683  df-in 3072  df-ss 3079  df-dcin 12990 This theorem is referenced by:  sumdc2  12995
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