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Theorem nfwe 4092
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
nfwe.r 𝑥𝑅
nfwe.a 𝑥𝐴
Assertion
Ref Expression
nfwe 𝑥 𝑅 We 𝐴

Proof of Theorem nfwe
Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-wetr 4071 . 2 (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ ∀𝑎𝐴𝑏𝐴𝑐𝐴 ((𝑎𝑅𝑏𝑏𝑅𝑐) → 𝑎𝑅𝑐)))
2 nfwe.r . . . 4 𝑥𝑅
3 nfwe.a . . . 4 𝑥𝐴
42, 3nffr 4086 . . 3 𝑥 𝑅 Fr 𝐴
5 nfcv 2178 . . . . . . . . 9 𝑥𝑎
6 nfcv 2178 . . . . . . . . 9 𝑥𝑏
75, 2, 6nfbr 3808 . . . . . . . 8 𝑥 𝑎𝑅𝑏
8 nfcv 2178 . . . . . . . . 9 𝑥𝑐
96, 2, 8nfbr 3808 . . . . . . . 8 𝑥 𝑏𝑅𝑐
107, 9nfan 1457 . . . . . . 7 𝑥(𝑎𝑅𝑏𝑏𝑅𝑐)
115, 2, 8nfbr 3808 . . . . . . 7 𝑥 𝑎𝑅𝑐
1210, 11nfim 1464 . . . . . 6 𝑥((𝑎𝑅𝑏𝑏𝑅𝑐) → 𝑎𝑅𝑐)
133, 12nfralxy 2360 . . . . 5 𝑥𝑐𝐴 ((𝑎𝑅𝑏𝑏𝑅𝑐) → 𝑎𝑅𝑐)
143, 13nfralxy 2360 . . . 4 𝑥𝑏𝐴𝑐𝐴 ((𝑎𝑅𝑏𝑏𝑅𝑐) → 𝑎𝑅𝑐)
153, 14nfralxy 2360 . . 3 𝑥𝑎𝐴𝑏𝐴𝑐𝐴 ((𝑎𝑅𝑏𝑏𝑅𝑐) → 𝑎𝑅𝑐)
164, 15nfan 1457 . 2 𝑥(𝑅 Fr 𝐴 ∧ ∀𝑎𝐴𝑏𝐴𝑐𝐴 ((𝑎𝑅𝑏𝑏𝑅𝑐) → 𝑎𝑅𝑐))
171, 16nfxfr 1363 1 𝑥 𝑅 We 𝐴
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wnf 1349  wnfc 2165  wral 2306   class class class wbr 3764   Fr wfr 4065   We wwe 4067
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-frfor 4068  df-frind 4069  df-wetr 4071
This theorem is referenced by: (None)
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