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Theorem nfso 4120
Description: Bound-variable hypothesis builder for total orders. (Contributed by Stefan O'Rear, 20-Jan-2015.)
Hypotheses
Ref Expression
nfpo.r  |-  F/_ x R
nfpo.a  |-  F/_ x A
Assertion
Ref Expression
nfso  |-  F/ x  R  Or  A

Proof of Theorem nfso
Dummy variables  a  b  c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-iso 4115 . 2  |-  ( R  Or  A  <->  ( R  Po  A  /\  A. a  e.  A  A. b  e.  A  A. c  e.  A  ( a R b  ->  (
a R c  \/  c R b ) ) ) )
2 nfpo.r . . . 4  |-  F/_ x R
3 nfpo.a . . . 4  |-  F/_ x A
42, 3nfpo 4119 . . 3  |-  F/ x  R  Po  A
5 nfcv 2228 . . . . . . . 8  |-  F/_ x
a
6 nfcv 2228 . . . . . . . 8  |-  F/_ x
b
75, 2, 6nfbr 3881 . . . . . . 7  |-  F/ x  a R b
8 nfcv 2228 . . . . . . . . 9  |-  F/_ x
c
95, 2, 8nfbr 3881 . . . . . . . 8  |-  F/ x  a R c
108, 2, 6nfbr 3881 . . . . . . . 8  |-  F/ x  c R b
119, 10nfor 1511 . . . . . . 7  |-  F/ x
( a R c  \/  c R b )
127, 11nfim 1509 . . . . . 6  |-  F/ x
( a R b  ->  ( a R c  \/  c R b ) )
133, 12nfralxy 2414 . . . . 5  |-  F/ x A. c  e.  A  ( a R b  ->  ( a R c  \/  c R b ) )
143, 13nfralxy 2414 . . . 4  |-  F/ x A. b  e.  A  A. c  e.  A  ( a R b  ->  ( a R c  \/  c R b ) )
153, 14nfralxy 2414 . . 3  |-  F/ x A. a  e.  A  A. b  e.  A  A. c  e.  A  ( a R b  ->  ( a R c  \/  c R b ) )
164, 15nfan 1502 . 2  |-  F/ x
( R  Po  A  /\  A. a  e.  A  A. b  e.  A  A. c  e.  A  ( a R b  ->  ( a R c  \/  c R b ) ) )
171, 16nfxfr 1408 1  |-  F/ x  R  Or  A
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    \/ wo 664   F/wnf 1394   F/_wnfc 2215   A.wral 2359   class class class wbr 3837    Po wpo 4112    Or wor 4113
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621  df-un 3001  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-po 4114  df-iso 4115
This theorem is referenced by: (None)
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