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Theorem nfso 4509
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 4504 . 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 4508 . . 3  |-  F/ x  R  Po  A
5 nfcv 2572 . . . . . . 7  |-  F/_ x
a
6 nfcv 2572 . . . . . . 7  |-  F/_ x
b
75, 2, 6nfbr 4256 . . . . . 6  |-  F/ x  a R b
8 nfv 1629 . . . . . 6  |-  F/ x  a  =  b
96, 2, 5nfbr 4256 . . . . . 6  |-  F/ x  b R a
107, 8, 9nf3or 1859 . . . . 5  |-  F/ x
( a R b  \/  a  =  b  \/  b R a )
113, 10nfral 2759 . . . 4  |-  F/ x A. b  e.  A  ( a R b  \/  a  =  b  \/  b R a )
123, 11nfral 2759 . . 3  |-  F/ x A. a  e.  A  A. b  e.  A  ( a R b  \/  a  =  b  \/  b R a )
134, 12nfan 1846 . 2  |-  F/ x
( R  Po  A  /\  A. a  e.  A  A. b  e.  A  ( a R b  \/  a  =  b  \/  b R a ) )
141, 13nfxfr 1579 1  |-  F/ x  R  Or  A
Colors of variables: wff set class
Syntax hints:    /\ wa 359    \/ w3o 935   F/wnf 1553   F/_wnfc 2559   A.wral 2705   class class class wbr 4212    Po wpo 4501    Or wor 4502
This theorem is referenced by:  nfwe  4558
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-po 4503  df-so 4504
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