Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  dffr2 Unicode version

Theorem dffr2 4539
 Description: Alternate definition of well-founded relation. Similar to Definition 6.21 of [TakeutiZaring] p. 30. (Contributed by NM, 17-Feb-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Proof shortened by Mario Carneiro, 23-Jun-2015.)
Assertion
Ref Expression
dffr2
Distinct variable groups:   ,,,   ,,,

Proof of Theorem dffr2
StepHypRef Expression
1 df-fr 4533 . 2
2 rabeq0 3641 . . . . 5
32rexbii 2722 . . . 4
43imbi2i 304 . . 3
54albii 1575 . 2
61, 5bitr4i 244 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 177   wa 359  wal 1549   wceq 1652   wne 2598  wral 2697  wrex 2698  crab 2701   wss 3312  c0 3620   class class class wbr 4204   wfr 4530 This theorem is referenced by:  fr0  4553  dfepfr  4559  dffr3  5227 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-nul 3621  df-fr 4533
 Copyright terms: Public domain W3C validator