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Theorem nfso 4320
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 4315 . 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 4319 . . 3  |-  F/ x  R  Po  A
5 nfcv 2332 . . . . . . . 8  |-  F/_ x
a
6 nfcv 2332 . . . . . . . 8  |-  F/_ x
b
75, 2, 6nfbr 4064 . . . . . . 7  |-  F/ x  a R b
8 nfcv 2332 . . . . . . . . 9  |-  F/_ x
c
95, 2, 8nfbr 4064 . . . . . . . 8  |-  F/ x  a R c
108, 2, 6nfbr 4064 . . . . . . . 8  |-  F/ x  c R b
119, 10nfor 1585 . . . . . . 7  |-  F/ x
( a R c  \/  c R b )
127, 11nfim 1583 . . . . . 6  |-  F/ x
( a R b  ->  ( a R c  \/  c R b ) )
133, 12nfralxy 2528 . . . . 5  |-  F/ x A. c  e.  A  ( a R b  ->  ( a R c  \/  c R b ) )
143, 13nfralxy 2528 . . . 4  |-  F/ x A. b  e.  A  A. c  e.  A  ( a R b  ->  ( a R c  \/  c R b ) )
153, 14nfralxy 2528 . . 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 1576 . 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 1485 1  |-  F/ x  R  Or  A
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    \/ wo 709   F/wnf 1471   F/_wnfc 2319   A.wral 2468   class class class wbr 4018    Po wpo 4312    Or wor 4313
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-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-v 2754  df-un 3148  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-po 4314  df-iso 4315
This theorem is referenced by: (None)
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