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Theorem nfwe 4367
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 4346 . 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 4361 . . 3  |-  F/ x  R  Fr  A
5 nfcv 2329 . . . . . . . . 9  |-  F/_ x
a
6 nfcv 2329 . . . . . . . . 9  |-  F/_ x
b
75, 2, 6nfbr 4061 . . . . . . . 8  |-  F/ x  a R b
8 nfcv 2329 . . . . . . . . 9  |-  F/_ x
c
96, 2, 8nfbr 4061 . . . . . . . 8  |-  F/ x  b R c
107, 9nfan 1575 . . . . . . 7  |-  F/ x
( a R b  /\  b R c )
115, 2, 8nfbr 4061 . . . . . . 7  |-  F/ x  a R c
1210, 11nfim 1582 . . . . . 6  |-  F/ x
( ( a R b  /\  b R c )  ->  a R c )
133, 12nfralxy 2525 . . . . 5  |-  F/ x A. c  e.  A  ( ( a R b  /\  b R c )  ->  a R c )
143, 13nfralxy 2525 . . . 4  |-  F/ x A. b  e.  A  A. c  e.  A  ( ( a R b  /\  b R c )  ->  a R c )
153, 14nfralxy 2525 . . 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 1575 . 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 1484 1  |-  F/ x  R  We  A
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104   F/wnf 1470   F/_wnfc 2316   A.wral 2465   class class class wbr 4015    Fr wfr 4340    We wwe 4342
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-sn 3610  df-pr 3611  df-op 3613  df-br 4016  df-frfor 4343  df-frind 4344  df-wetr 4346
This theorem is referenced by: (None)
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