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Theorem nfwe 5648
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 5629 . 2 (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴𝑅 Or 𝐴))
2 nffr.r . . . 4 𝑥𝑅
3 nffr.a . . . 4 𝑥𝐴
42, 3nffr 5646 . . 3 𝑥 𝑅 Fr 𝐴
52, 3nfso 5590 . . 3 𝑥 𝑅 Or 𝐴
64, 5nfan 1903 . 2 𝑥(𝑅 Fr 𝐴𝑅 Or 𝐴)
71, 6nfxfr 1856 1 𝑥 𝑅 We 𝐴
Colors of variables: wff setvar class
Syntax hints:  wa 397  wnf 1786  wnfc 2884   Or wor 5583   Fr wfr 5624   We wwe 5626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5145  df-po 5584  df-so 5585  df-fr 5627  df-we 5629
This theorem is referenced by:  nfoi  9496  aomclem6  41672
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