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Theorem nfwe 5242
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 5227 . 2 (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴𝑅 Or 𝐴))
2 nffr.r . . . 4 𝑥𝑅
3 nffr.a . . . 4 𝑥𝐴
42, 3nffr 5240 . . 3 𝑥 𝑅 Fr 𝐴
52, 3nfso 5193 . . 3 𝑥 𝑅 Or 𝐴
64, 5nfan 1977 . 2 𝑥(𝑅 Fr 𝐴𝑅 Or 𝐴)
71, 6nfxfr 1928 1 𝑥 𝑅 We 𝐴
Colors of variables: wff setvar class
Syntax hints:  wa 383  wnf 1857  wnfc 2889   Or wor 5186   Fr wfr 5222   We wwe 5224
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-po 5187  df-so 5188  df-fr 5225  df-we 5227
This theorem is referenced by:  nfoi  8584  aomclem6  38131
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