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Theorem isomni 7001
 Description: The predicate of being omniscient. (Contributed by Jim Kingdon, 28-Jun-2022.)
Assertion
Ref Expression
isomni Omni
Distinct variable group:   ,,
Allowed substitution hints:   (,)

Proof of Theorem isomni
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 feq2 5251 . . . 4
2 rexeq 2625 . . . . 5
3 raleq 2624 . . . . 5
42, 3orbi12d 782 . . . 4
51, 4imbi12d 233 . . 3
65albidv 1796 . 2
7 df-omni 6999 . 2 Omni
86, 7elab2g 2826 1 Omni
 Colors of variables: wff set class Syntax hints:   wi 4   wb 104   wo 697  wal 1329   wceq 1331   wcel 1480  wral 2414  wrex 2415  c0 3358  wf 5114  cfv 5118  c1o 6299  c2o 6300  Omnicomni 6997 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 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-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-fn 5121  df-f 5122  df-omni 6999 This theorem is referenced by:  isomnimap  7002  finomni  7005  exmidomniim  7006  exmidomni  7007  omnimkv  7023
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