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Theorem nfwe 4247
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 4226 . 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 4241 . . 3  |-  F/ x  R  Fr  A
5 nfcv 2258 . . . . . . . . 9  |-  F/_ x
a
6 nfcv 2258 . . . . . . . . 9  |-  F/_ x
b
75, 2, 6nfbr 3944 . . . . . . . 8  |-  F/ x  a R b
8 nfcv 2258 . . . . . . . . 9  |-  F/_ x
c
96, 2, 8nfbr 3944 . . . . . . . 8  |-  F/ x  b R c
107, 9nfan 1529 . . . . . . 7  |-  F/ x
( a R b  /\  b R c )
115, 2, 8nfbr 3944 . . . . . . 7  |-  F/ x  a R c
1210, 11nfim 1536 . . . . . 6  |-  F/ x
( ( a R b  /\  b R c )  ->  a R c )
133, 12nfralxy 2448 . . . . 5  |-  F/ x A. c  e.  A  ( ( a R b  /\  b R c )  ->  a R c )
143, 13nfralxy 2448 . . . 4  |-  F/ x A. b  e.  A  A. c  e.  A  ( ( a R b  /\  b R c )  ->  a R c )
153, 14nfralxy 2448 . . 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 1529 . 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 1435 1  |-  F/ x  R  We  A
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   F/wnf 1421   F/_wnfc 2245   A.wral 2393   class class class wbr 3899    Fr wfr 4220    We wwe 4222
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-frfor 4223  df-frind 4224  df-wetr 4226
This theorem is referenced by: (None)
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