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Theorem nfso 4795
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 4790 . 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 4794 . . 3  |-  F/ x  R  Po  A
5 nfcv 2616 . . . . . . 7  |-  F/_ x
a
6 nfcv 2616 . . . . . . 7  |-  F/_ x
b
75, 2, 6nfbr 4483 . . . . . 6  |-  F/ x  a R b
8 nfv 1712 . . . . . 6  |-  F/ x  a  =  b
96, 2, 5nfbr 4483 . . . . . 6  |-  F/ x  b R a
107, 8, 9nf3or 1941 . . . . 5  |-  F/ x
( a R b  \/  a  =  b  \/  b R a )
113, 10nfral 2840 . . . 4  |-  F/ x A. b  e.  A  ( a R b  \/  a  =  b  \/  b R a )
123, 11nfral 2840 . . 3  |-  F/ x A. a  e.  A  A. b  e.  A  ( a R b  \/  a  =  b  \/  b R a )
134, 12nfan 1933 . 2  |-  F/ x
( R  Po  A  /\  A. a  e.  A  A. b  e.  A  ( a R b  \/  a  =  b  \/  b R a ) )
141, 13nfxfr 1650 1  |-  F/ x  R  Or  A
Colors of variables: wff setvar class
Syntax hints:    /\ wa 367    \/ w3o 970   F/wnf 1621   F/_wnfc 2602   A.wral 2804   class class class wbr 4439    Po wpo 4787    Or wor 4788
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-po 4789  df-so 4790
This theorem is referenced by:  nfwe  4844
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