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Theorem dcextest 4463
 Description: If it is decidable whether is a set, then is decidable (where does not occur in ). From this fact, we can deduce (outside the formal system, since we cannot quantify over classes) that if it is decidable whether any class is a set, then "weak excluded middle" (that is, any negated proposition is decidable) holds. (Contributed by Jim Kingdon, 3-Jul-2022.)
Hypothesis
Ref Expression
dcextest.ex DECID
Assertion
Ref Expression
dcextest DECID
Distinct variable group:   ,

Proof of Theorem dcextest
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 dcextest.ex . . . 4 DECID
2 exmiddc 804 . . . 4 DECID
31, 2ax-mp 5 . . 3
4 vprc 4028 . . . . . . 7
5 df-v 2660 . . . . . . . . 9
6 equid 1660 . . . . . . . . . . 11
7 pm5.1im 172 . . . . . . . . . . 11
86, 7ax-mp 5 . . . . . . . . . 10
98abbidv 2233 . . . . . . . . 9
105, 9syl5req 2161 . . . . . . . 8
1110eleq1d 2184 . . . . . . 7
124, 11mtbiri 647 . . . . . 6
1312con2i 599 . . . . 5
14 vex 2661 . . . . . . . . . 10
15 biidd 171 . . . . . . . . . 10
1614, 15elab 2800 . . . . . . . . 9
1716notbii 640 . . . . . . . 8
1817biimpri 132 . . . . . . 7
1918eq0rdv 3375 . . . . . 6
20 0ex 4023 . . . . . 6
2119, 20syl6eqel 2206 . . . . 5
2213, 21impbii 125 . . . 4
2322notbii 640 . . . 4
2422, 23orbi12i 736 . . 3
253, 24mpbi 144 . 2
26 df-dc 803 . 2 DECID
2725, 26mpbir 145 1 DECID
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 104   wo 680  DECID wdc 802   wcel 1463  cab 2101  cvv 2658  c0 3331 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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-nul 4022 This theorem depends on definitions:  df-bi 116  df-dc 803  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-dif 3041  df-in 3045  df-ss 3052  df-nul 3332 This theorem is referenced by: (None)
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