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Theorem ifmdc 3542
 Description: If a conditional class is inhabited, then the condition is decidable. This shows that conditionals are not very useful unless one can prove the condition decidable. (Contributed by BJ, 24-Sep-2022.)
Assertion
Ref Expression
ifmdc DECID

Proof of Theorem ifmdc
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eleq1 2220 . . . . 5
21imbi1d 230 . . . 4
3 df-if 3506 . . . . . 6
43abeq2i 2268 . . . . 5
5 simpr 109 . . . . . 6
6 simpr 109 . . . . . 6
75, 6orim12i 749 . . . . 5
84, 7sylbi 120 . . . 4
92, 8vtoclg 2772 . . 3
109pm2.43i 49 . 2
11 df-dc 821 . 2 DECID
1210, 11sylibr 133 1 DECID
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 103   wo 698  DECID wdc 820   wceq 1335   wcel 2128  cif 3505 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-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139 This theorem depends on definitions:  df-bi 116  df-dc 821  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-if 3506 This theorem is referenced by: (None)
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