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Theorem nfwe 4827
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 4812 . 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 4825 . . 3  |-  F/ x  R  Fr  A
52, 3nfso 4778 . . 3  |-  F/ x  R  Or  A
64, 5nfan 1985 . 2  |-  F/ x
( R  Fr  A  /\  R  Or  A
)
71, 6nfxfr 1693 1  |-  F/ x  R  We  A
Colors of variables: wff setvar class
Syntax hints:    /\ wa 371   F/wnf 1664   F/_wnfc 2571    Or wor 4771    Fr wfr 4807    We wwe 4809
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ral 2781  df-rex 2782  df-rab 2785  df-v 3084  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-sn 3998  df-pr 4000  df-op 4004  df-br 4422  df-po 4772  df-so 4773  df-fr 4810  df-we 4812
This theorem is referenced by:  nfoi  8033  aomclem6  35881
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