ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nfso Unicode version

Theorem nfso 4219
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 4214 . 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 4218 . . 3  |-  F/ x  R  Po  A
5 nfcv 2279 . . . . . . . 8  |-  F/_ x
a
6 nfcv 2279 . . . . . . . 8  |-  F/_ x
b
75, 2, 6nfbr 3969 . . . . . . 7  |-  F/ x  a R b
8 nfcv 2279 . . . . . . . . 9  |-  F/_ x
c
95, 2, 8nfbr 3969 . . . . . . . 8  |-  F/ x  a R c
108, 2, 6nfbr 3969 . . . . . . . 8  |-  F/ x  c R b
119, 10nfor 1553 . . . . . . 7  |-  F/ x
( a R c  \/  c R b )
127, 11nfim 1551 . . . . . 6  |-  F/ x
( a R b  ->  ( a R c  \/  c R b ) )
133, 12nfralxy 2469 . . . . 5  |-  F/ x A. c  e.  A  ( a R b  ->  ( a R c  \/  c R b ) )
143, 13nfralxy 2469 . . . 4  |-  F/ x A. b  e.  A  A. c  e.  A  ( a R b  ->  ( a R c  \/  c R b ) )
153, 14nfralxy 2469 . . 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 1544 . 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 1450 1  |-  F/ x  R  Or  A
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    \/ wo 697   F/wnf 1436   F/_wnfc 2266   A.wral 2414   class class class wbr 3924    Po wpo 4211    Or wor 4212
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-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-v 2683  df-un 3070  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-po 4213  df-iso 4214
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator