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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 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-so 4315 . 2  |-  ( R  Or  A  <->  ( R  Po  A  /\  A. a  e.  A  A. b  e.  A  ( a R b  \/  a  =  b  \/  b R a ) ) )
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 2419 . . . . . . 7  |-  F/_ x
a
6 nfcv 2419 . . . . . . 7  |-  F/_ x
b
75, 2, 6nfbr 4067 . . . . . 6  |-  F/ x  a R b
8 nfv 1605 . . . . . 6  |-  F/ x  a  =  b
96, 2, 5nfbr 4067 . . . . . 6  |-  F/ x  b R a
107, 8, 9nf3or 1773 . . . . 5  |-  F/ x
( a R b  \/  a  =  b  \/  b R a )
113, 10nfral 2596 . . . 4  |-  F/ x A. b  e.  A  ( a R b  \/  a  =  b  \/  b R a )
123, 11nfral 2596 . . 3  |-  F/ x A. a  e.  A  A. b  e.  A  ( a R b  \/  a  =  b  \/  b R a )
134, 12nfan 1771 . 2  |-  F/ x
( R  Po  A  /\  A. a  e.  A  A. b  e.  A  ( a R b  \/  a  =  b  \/  b R a ) )
141, 13nfxfr 1557 1  |-  F/ x  R  Or  A
Colors of variables: wff set class
Syntax hints:    /\ wa 358    \/ w3o 933   F/wnf 1531    = wceq 1623   F/_wnfc 2406   A.wral 2543   class class class wbr 4023    Po wpo 4312    Or wor 4313
This theorem is referenced by:  nfwe  4369
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-po 4314  df-so 4315
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