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Theorem nfwe 5496
 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 𝑥𝑅
nffr.a 𝑥𝐴
Assertion
Ref Expression
nfwe 𝑥 𝑅 We 𝐴

Proof of Theorem nfwe
StepHypRef Expression
1 df-we 5481 . 2 (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴𝑅 Or 𝐴))
2 nffr.r . . . 4 𝑥𝑅
3 nffr.a . . . 4 𝑥𝐴
42, 3nffr 5494 . . 3 𝑥 𝑅 Fr 𝐴
52, 3nfso 5445 . . 3 𝑥 𝑅 Or 𝐴
64, 5nfan 1900 . 2 𝑥(𝑅 Fr 𝐴𝑅 Or 𝐴)
71, 6nfxfr 1854 1 𝑥 𝑅 We 𝐴
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 399  Ⅎwnf 1785  Ⅎwnfc 2936   Or wor 5438   Fr wfr 5476   We wwe 5478 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5032  df-po 5439  df-so 5440  df-fr 5479  df-we 5481 This theorem is referenced by:  nfoi  8965  aomclem6  40046
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