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Theorem nfwe 5660
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 5639 . 2 (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴𝑅 Or 𝐴))
2 nffr.r . . . 4 𝑥𝑅
3 nffr.a . . . 4 𝑥𝐴
42, 3nffr 5658 . . 3 𝑥 𝑅 Fr 𝐴
52, 3nfso 5599 . . 3 𝑥 𝑅 Or 𝐴
64, 5nfan 1899 . 2 𝑥(𝑅 Fr 𝐴𝑅 Or 𝐴)
71, 6nfxfr 1853 1 𝑥 𝑅 We 𝐴
Colors of variables: wff setvar class
Syntax hints:  wa 395  wnf 1783  wnfc 2890   Or wor 5591   Fr wfr 5634   We wwe 5636
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-po 5592  df-so 5593  df-fr 5637  df-we 5639
This theorem is referenced by:  nfoi  9554  aomclem6  43071
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