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Theorem nfwe 5527
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 5511 . 2 (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴𝑅 Or 𝐴))
2 nffr.r . . . 4 𝑥𝑅
3 nffr.a . . . 4 𝑥𝐴
42, 3nffr 5525 . . 3 𝑥 𝑅 Fr 𝐴
52, 3nfso 5474 . . 3 𝑥 𝑅 Or 𝐴
64, 5nfan 1907 . 2 𝑥(𝑅 Fr 𝐴𝑅 Or 𝐴)
71, 6nfxfr 1860 1 𝑥 𝑅 We 𝐴
Colors of variables: wff setvar class
Syntax hints:  wa 399  wnf 1791  wnfc 2884   Or wor 5467   Fr wfr 5506   We wwe 5508
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-br 5054  df-po 5468  df-so 5469  df-fr 5509  df-we 5511
This theorem is referenced by:  nfoi  9130  aomclem6  40587
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