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

Theorem nffr 4559
 Description: Bound-variable hypothesis builder for well-founded relations. (Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Mario Carneiro, 14-Oct-2016.)
Hypotheses
Ref Expression
nffr.r
nffr.a
Assertion
Ref Expression
nffr

Proof of Theorem nffr
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-fr 4544 . 2
2 nfcv 2574 . . . . . 6
3 nffr.a . . . . . 6
42, 3nfss 3343 . . . . 5
5 nfv 1630 . . . . 5
64, 5nfan 1847 . . . 4
7 nfcv 2574 . . . . . . . 8
8 nffr.r . . . . . . . 8
9 nfcv 2574 . . . . . . . 8
107, 8, 9nfbr 4259 . . . . . . 7
1110nfn 1812 . . . . . 6
122, 11nfral 2761 . . . . 5
132, 12nfrex 2763 . . . 4
146, 13nfim 1833 . . 3
1514nfal 1865 . 2
161, 15nfxfr 1580 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 360  wal 1550  wnf 1554  wnfc 2561   wne 2601  wral 2707  wrex 2708   wss 3322  c0 3630   class class class wbr 4215   wfr 4541 This theorem is referenced by:  nfwe  4561 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-fr 4544
 Copyright terms: Public domain W3C validator