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Theorem nfwe 4769
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 4754 . 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 4767 . . 3  |-  F/ x  R  Fr  A
52, 3nfso 4720 . . 3  |-  F/ x  R  Or  A
64, 5nfan 1936 . 2  |-  F/ x
( R  Fr  A  /\  R  Or  A
)
71, 6nfxfr 1653 1  |-  F/ x  R  We  A
Colors of variables: wff setvar class
Syntax hints:    /\ wa 367   F/wnf 1624   F/_wnfc 2530    Or wor 4713    Fr wfr 4749    We wwe 4751
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-br 4368  df-po 4714  df-so 4715  df-fr 4752  df-we 4754
This theorem is referenced by:  nfoi  7854  aomclem6  31171
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