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Theorem nfpo 4749
Description: Bound-variable hypothesis builder for partial 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
nfpo  |-  F/ x  R  Po  A

Proof of Theorem nfpo
Dummy variables  a 
b  c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-po 4744 . 2  |-  ( R  Po  A  <->  A. a  e.  A  A. b  e.  A  A. c  e.  A  ( -.  a R a  /\  (
( a R b  /\  b R c )  ->  a R
c ) ) )
2 nfpo.a . . 3  |-  F/_ x A
3 nfcv 2614 . . . . . . . 8  |-  F/_ x
a
4 nfpo.r . . . . . . . 8  |-  F/_ x R
53, 4, 3nfbr 4439 . . . . . . 7  |-  F/ x  a R a
65nfn 1839 . . . . . 6  |-  F/ x  -.  a R a
7 nfcv 2614 . . . . . . . . 9  |-  F/_ x
b
83, 4, 7nfbr 4439 . . . . . . . 8  |-  F/ x  a R b
9 nfcv 2614 . . . . . . . . 9  |-  F/_ x
c
107, 4, 9nfbr 4439 . . . . . . . 8  |-  F/ x  b R c
118, 10nfan 1865 . . . . . . 7  |-  F/ x
( a R b  /\  b R c )
123, 4, 9nfbr 4439 . . . . . . 7  |-  F/ x  a R c
1311, 12nfim 1857 . . . . . 6  |-  F/ x
( ( a R b  /\  b R c )  ->  a R c )
146, 13nfan 1865 . . . . 5  |-  F/ x
( -.  a R a  /\  ( ( a R b  /\  b R c )  -> 
a R c ) )
152, 14nfral 2882 . . . 4  |-  F/ x A. c  e.  A  ( -.  a R
a  /\  ( (
a R b  /\  b R c )  -> 
a R c ) )
162, 15nfral 2882 . . 3  |-  F/ x A. b  e.  A  A. c  e.  A  ( -.  a R
a  /\  ( (
a R b  /\  b R c )  -> 
a R c ) )
172, 16nfral 2882 . 2  |-  F/ x A. a  e.  A  A. b  e.  A  A. c  e.  A  ( -.  a R
a  /\  ( (
a R b  /\  b R c )  -> 
a R c ) )
181, 17nfxfr 1616 1  |-  F/ x  R  Po  A
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369   F/wnf 1590   F/_wnfc 2600   A.wral 2796   class class class wbr 4395    Po wpo 4742
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-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
This theorem is referenced by:  nfso  4750
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