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Theorem exmid1dc 4132
 Description: A convenience theorem for proving that something implies EXMID. Think of this as an alternative to using a proposition, as in proofs like undifexmid 4126 or ordtriexmid 4446. In this context can be thought of as "x is true". (Contributed by Jim Kingdon, 21-Nov-2023.)
Hypothesis
Ref Expression
exmid1dc.x DECID
Assertion
Ref Expression
exmid1dc EXMID
Distinct variable group:   ,

Proof of Theorem exmid1dc
StepHypRef Expression
1 exmid1dc.x . . . . . . 7 DECID
2 exmiddc 822 . . . . . . 7 DECID
31, 2syl 14 . . . . . 6
4 df-ne 2310 . . . . . . . . 9
5 pwntru 4131 . . . . . . . . . 10
65ex 114 . . . . . . . . 9
74, 6syl5bir 152 . . . . . . . 8
87orim2d 778 . . . . . . 7
98adantl 275 . . . . . 6
103, 9mpd 13 . . . . 5
1110orcomd 719 . . . 4
1211ex 114 . . 3
1312alrimiv 1847 . 2
14 exmid01 4130 . 2 EXMID
1513, 14sylibr 133 1 EXMID
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 103   wo 698  DECID wdc 820  wal 1330   wceq 1332   wne 2309   wss 3077  c0 3369  csn 3533  EXMIDwem 4127 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 604  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-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-nul 4063 This theorem depends on definitions:  df-bi 116  df-dc 821  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-v 2692  df-dif 3079  df-in 3083  df-ss 3090  df-nul 3370  df-sn 3539  df-exmid 4128 This theorem is referenced by:  exmidonfin  7070  exmidaclem  7084
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