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

Theorem nfwe 4815
Description: Bound-variable hypothesis builder for well-orderings. (Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Mario Carneiro, 14-Oct-2016.)
Hypotheses
Ref Expression
nffr.r  |-  F/_ x R
nffr.a  |-  F/_ x A
Assertion
Ref Expression
nfwe  |-  F/ x  R  We  A

Proof of Theorem nfwe
StepHypRef Expression
1 df-we 4800 . 2  |-  ( R  We  A  <->  ( R  Fr  A  /\  R  Or  A ) )
2 nffr.r . . . 4  |-  F/_ x R
3 nffr.a . . . 4  |-  F/_ x A
42, 3nffr 4813 . . 3  |-  F/ x  R  Fr  A
52, 3nfso 4766 . . 3  |-  F/ x  R  Or  A
64, 5nfan 2031 . 2  |-  F/ x
( R  Fr  A  /\  R  Or  A
)
71, 6nfxfr 1704 1  |-  F/ x  R  We  A
Colors of variables: wff setvar class
Syntax hints:    /\ wa 376   F/wnf 1675   F/_wnfc 2599    Or wor 4759    Fr wfr 4795    We wwe 4797
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-po 4760  df-so 4761  df-fr 4798  df-we 4800
This theorem is referenced by:  nfoi  8047  aomclem6  35988
  Copyright terms: Public domain W3C validator