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Theorem nfwe 4277
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 4256 . 2 (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ ∀𝑎𝐴𝑏𝐴𝑐𝐴 ((𝑎𝑅𝑏𝑏𝑅𝑐) → 𝑎𝑅𝑐)))
2 nfwe.r . . . 4 𝑥𝑅
3 nfwe.a . . . 4 𝑥𝐴
42, 3nffr 4271 . . 3 𝑥 𝑅 Fr 𝐴
5 nfcv 2281 . . . . . . . . 9 𝑥𝑎
6 nfcv 2281 . . . . . . . . 9 𝑥𝑏
75, 2, 6nfbr 3974 . . . . . . . 8 𝑥 𝑎𝑅𝑏
8 nfcv 2281 . . . . . . . . 9 𝑥𝑐
96, 2, 8nfbr 3974 . . . . . . . 8 𝑥 𝑏𝑅𝑐
107, 9nfan 1544 . . . . . . 7 𝑥(𝑎𝑅𝑏𝑏𝑅𝑐)
115, 2, 8nfbr 3974 . . . . . . 7 𝑥 𝑎𝑅𝑐
1210, 11nfim 1551 . . . . . 6 𝑥((𝑎𝑅𝑏𝑏𝑅𝑐) → 𝑎𝑅𝑐)
133, 12nfralxy 2471 . . . . 5 𝑥𝑐𝐴 ((𝑎𝑅𝑏𝑏𝑅𝑐) → 𝑎𝑅𝑐)
143, 13nfralxy 2471 . . . 4 𝑥𝑏𝐴𝑐𝐴 ((𝑎𝑅𝑏𝑏𝑅𝑐) → 𝑎𝑅𝑐)
153, 14nfralxy 2471 . . 3 𝑥𝑎𝐴𝑏𝐴𝑐𝐴 ((𝑎𝑅𝑏𝑏𝑅𝑐) → 𝑎𝑅𝑐)
164, 15nfan 1544 . 2 𝑥(𝑅 Fr 𝐴 ∧ ∀𝑎𝐴𝑏𝐴𝑐𝐴 ((𝑎𝑅𝑏𝑏𝑅𝑐) → 𝑎𝑅𝑐))
171, 16nfxfr 1450 1 𝑥 𝑅 We 𝐴
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wnf 1436  wnfc 2268  wral 2416   class class class wbr 3929   Fr wfr 4250   We wwe 4252
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-frfor 4253  df-frind 4254  df-wetr 4256
This theorem is referenced by: (None)
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