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Theorem nfso 4336
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 4331 . 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 4335 . . 3  |-  F/ x  R  Po  A
5 nfcv 2432 . . . . . . 7  |-  F/_ x
a
6 nfcv 2432 . . . . . . 7  |-  F/_ x
b
75, 2, 6nfbr 4083 . . . . . 6  |-  F/ x  a R b
8 nfv 1609 . . . . . 6  |-  F/ x  a  =  b
96, 2, 5nfbr 4083 . . . . . 6  |-  F/ x  b R a
107, 8, 9nf3or 1785 . . . . 5  |-  F/ x
( a R b  \/  a  =  b  \/  b R a )
113, 10nfral 2609 . . . 4  |-  F/ x A. b  e.  A  ( a R b  \/  a  =  b  \/  b R a )
123, 11nfral 2609 . . 3  |-  F/ x A. a  e.  A  A. b  e.  A  ( a R b  \/  a  =  b  \/  b R a )
134, 12nfan 1783 . 2  |-  F/ x
( R  Po  A  /\  A. a  e.  A  A. b  e.  A  ( a R b  \/  a  =  b  \/  b R a ) )
141, 13nfxfr 1560 1  |-  F/ x  R  Or  A
Colors of variables: wff set class
Syntax hints:    /\ wa 358    \/ w3o 933   F/wnf 1534    = wceq 1632   F/_wnfc 2419   A.wral 2556   class class class wbr 4039    Po wpo 4328    Or wor 4329
This theorem is referenced by:  nfwe  4385
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
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 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-po 4330  df-so 4331
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