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Theorem nfpo 4794
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 4789 . 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 2616 . . . . . . . 8  |-  F/_ x
a
4 nfpo.r . . . . . . . 8  |-  F/_ x R
53, 4, 3nfbr 4483 . . . . . . 7  |-  F/ x  a R a
65nfn 1906 . . . . . 6  |-  F/ x  -.  a R a
7 nfcv 2616 . . . . . . . . 9  |-  F/_ x
b
83, 4, 7nfbr 4483 . . . . . . . 8  |-  F/ x  a R b
9 nfcv 2616 . . . . . . . . 9  |-  F/_ x
c
107, 4, 9nfbr 4483 . . . . . . . 8  |-  F/ x  b R c
118, 10nfan 1933 . . . . . . 7  |-  F/ x
( a R b  /\  b R c )
123, 4, 9nfbr 4483 . . . . . . 7  |-  F/ x  a R c
1311, 12nfim 1925 . . . . . 6  |-  F/ x
( ( a R b  /\  b R c )  ->  a R c )
146, 13nfan 1933 . . . . 5  |-  F/ x
( -.  a R a  /\  ( ( a R b  /\  b R c )  -> 
a R c ) )
152, 14nfral 2840 . . . 4  |-  F/ x A. c  e.  A  ( -.  a R
a  /\  ( (
a R b  /\  b R c )  -> 
a R c ) )
162, 15nfral 2840 . . 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 2840 . 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 1650 1  |-  F/ x  R  Po  A
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367   F/wnf 1621   F/_wnfc 2602   A.wral 2804   class class class wbr 4439    Po wpo 4787
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-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
This theorem is referenced by:  nfso  4795
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