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Theorem nfwe 4191
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  |-  F/_ x R
nfwe.a  |-  F/_ x A
Assertion
Ref Expression
nfwe  |-  F/ 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 4170 . 2  |-  ( R  We  A  <->  ( R  Fr  A  /\  A. a  e.  A  A. b  e.  A  A. c  e.  A  ( (
a R b  /\  b R c )  -> 
a R c ) ) )
2 nfwe.r . . . 4  |-  F/_ x R
3 nfwe.a . . . 4  |-  F/_ x A
42, 3nffr 4185 . . 3  |-  F/ x  R  Fr  A
5 nfcv 2229 . . . . . . . . 9  |-  F/_ x
a
6 nfcv 2229 . . . . . . . . 9  |-  F/_ x
b
75, 2, 6nfbr 3895 . . . . . . . 8  |-  F/ x  a R b
8 nfcv 2229 . . . . . . . . 9  |-  F/_ x
c
96, 2, 8nfbr 3895 . . . . . . . 8  |-  F/ x  b R c
107, 9nfan 1503 . . . . . . 7  |-  F/ x
( a R b  /\  b R c )
115, 2, 8nfbr 3895 . . . . . . 7  |-  F/ x  a R c
1210, 11nfim 1510 . . . . . 6  |-  F/ x
( ( a R b  /\  b R c )  ->  a R c )
133, 12nfralxy 2415 . . . . 5  |-  F/ x A. c  e.  A  ( ( a R b  /\  b R c )  ->  a R c )
143, 13nfralxy 2415 . . . 4  |-  F/ x A. b  e.  A  A. c  e.  A  ( ( a R b  /\  b R c )  ->  a R c )
153, 14nfralxy 2415 . . 3  |-  F/ x A. a  e.  A  A. b  e.  A  A. c  e.  A  ( ( a R b  /\  b R c )  ->  a R c )
164, 15nfan 1503 . 2  |-  F/ x
( R  Fr  A  /\  A. a  e.  A  A. b  e.  A  A. c  e.  A  ( ( a R b  /\  b R c )  ->  a R c ) )
171, 16nfxfr 1409 1  |-  F/ x  R  We  A
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   F/wnf 1395   F/_wnfc 2216   A.wral 2360   class class class wbr 3851    Fr wfr 4164    We wwe 4166
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-sn 3456  df-pr 3457  df-op 3459  df-br 3852  df-frfor 4167  df-frind 4168  df-wetr 4170
This theorem is referenced by: (None)
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