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Theorem nfwe 4284
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 4263 . 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 4278 . . 3  |-  F/ x  R  Fr  A
5 nfcv 2282 . . . . . . . . 9  |-  F/_ x
a
6 nfcv 2282 . . . . . . . . 9  |-  F/_ x
b
75, 2, 6nfbr 3981 . . . . . . . 8  |-  F/ x  a R b
8 nfcv 2282 . . . . . . . . 9  |-  F/_ x
c
96, 2, 8nfbr 3981 . . . . . . . 8  |-  F/ x  b R c
107, 9nfan 1545 . . . . . . 7  |-  F/ x
( a R b  /\  b R c )
115, 2, 8nfbr 3981 . . . . . . 7  |-  F/ x  a R c
1210, 11nfim 1552 . . . . . 6  |-  F/ x
( ( a R b  /\  b R c )  ->  a R c )
133, 12nfralxy 2474 . . . . 5  |-  F/ x A. c  e.  A  ( ( a R b  /\  b R c )  ->  a R c )
143, 13nfralxy 2474 . . . 4  |-  F/ x A. b  e.  A  A. c  e.  A  ( ( a R b  /\  b R c )  ->  a R c )
153, 14nfralxy 2474 . . 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 1545 . 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 1451 1  |-  F/ x  R  We  A
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   F/wnf 1437   F/_wnfc 2269   A.wral 2417   class class class wbr 3936    Fr wfr 4257    We wwe 4259
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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-v 2691  df-un 3079  df-in 3081  df-ss 3088  df-sn 3537  df-pr 3538  df-op 3540  df-br 3937  df-frfor 4260  df-frind 4261  df-wetr 4263
This theorem is referenced by: (None)
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