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Theorem nfso 4750
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 4745 . 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 4749 . . 3  |-  F/ x  R  Po  A
5 nfcv 2614 . . . . . . 7  |-  F/_ x
a
6 nfcv 2614 . . . . . . 7  |-  F/_ x
b
75, 2, 6nfbr 4439 . . . . . 6  |-  F/ x  a R b
8 nfv 1674 . . . . . 6  |-  F/ x  a  =  b
96, 2, 5nfbr 4439 . . . . . 6  |-  F/ x  b R a
107, 8, 9nf3or 1873 . . . . 5  |-  F/ x
( a R b  \/  a  =  b  \/  b R a )
113, 10nfral 2882 . . . 4  |-  F/ x A. b  e.  A  ( a R b  \/  a  =  b  \/  b R a )
123, 11nfral 2882 . . 3  |-  F/ x A. a  e.  A  A. b  e.  A  ( a R b  \/  a  =  b  \/  b R a )
134, 12nfan 1865 . 2  |-  F/ x
( R  Po  A  /\  A. a  e.  A  A. b  e.  A  ( a R b  \/  a  =  b  \/  b R a ) )
141, 13nfxfr 1616 1  |-  F/ x  R  Or  A
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    \/ w3o 964   F/wnf 1590   F/_wnfc 2600   A.wral 2796   class class class wbr 4395    Po wpo 4742    Or wor 4743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ral 2801  df-rab 2805  df-v 3074  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-br 4396  df-po 4744  df-so 4745
This theorem is referenced by:  nfwe  4799
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