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Theorem fri 4785
 Description: Property of well-founded relation (one direction of definition). (Contributed by NM, 18-Mar-1997.)
Assertion
Ref Expression
fri
Distinct variable groups:   ,,   ,,   ,,
Allowed substitution hints:   (,)

Proof of Theorem fri
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 df-fr 4782 . . 3
2 sseq1 3463 . . . . . 6
3 neeq1 2684 . . . . . 6
42, 3anbi12d 709 . . . . 5
5 raleq 3004 . . . . . 6
65rexeqbi1dv 3013 . . . . 5
74, 6imbi12d 318 . . . 4
87spcgv 3144 . . 3
91, 8syl5bi 217 . 2
109imp31 430 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wa 367  wal 1403   wceq 1405   wcel 1842   wne 2598  wral 2754  wrex 2755   wss 3414  c0 3738   class class class wbr 4395   wfr 4779 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380 This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-v 3061  df-in 3421  df-ss 3428  df-fr 4782 This theorem is referenced by:  frc  4789  fr2nr  4801  frminex  4803  wereu  4819  wereu2  4820  fr3nr  6597  frfi  7799  fimax2g  7800  wofib  8004  wemapso  8010  wemapso2OLD  8011  wemapso2lem  8012  noinfep  8109  cflim2  8675  isfin1-3  8798  fin12  8825  fpwwe2lem12  9049  fpwwe2lem13  9050  fpwwe2  9051  bnj110  29243  frinfm  31508  fdc  31520  fnwe2lem2  35359
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