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Theorem nfpo 4450
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 4445 . 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 2524 . . . . . . . 8  |-  F/_ x
a
4 nfpo.r . . . . . . . 8  |-  F/_ x R
53, 4, 3nfbr 4198 . . . . . . 7  |-  F/ x  a R a
65nfn 1801 . . . . . 6  |-  F/ x  -.  a R a
7 nfcv 2524 . . . . . . . . 9  |-  F/_ x
b
83, 4, 7nfbr 4198 . . . . . . . 8  |-  F/ x  a R b
9 nfcv 2524 . . . . . . . . 9  |-  F/_ x
c
107, 4, 9nfbr 4198 . . . . . . . 8  |-  F/ x  b R c
118, 10nfan 1836 . . . . . . 7  |-  F/ x
( a R b  /\  b R c )
123, 4, 9nfbr 4198 . . . . . . 7  |-  F/ x  a R c
1311, 12nfim 1822 . . . . . 6  |-  F/ x
( ( a R b  /\  b R c )  ->  a R c )
146, 13nfan 1836 . . . . 5  |-  F/ x
( -.  a R a  /\  ( ( a R b  /\  b R c )  -> 
a R c ) )
152, 14nfral 2703 . . . 4  |-  F/ x A. c  e.  A  ( -.  a R
a  /\  ( (
a R b  /\  b R c )  -> 
a R c ) )
162, 15nfral 2703 . . 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 2703 . 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 1576 1  |-  F/ x  R  Po  A
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359   F/wnf 1550   F/_wnfc 2511   A.wral 2650   class class class wbr 4154    Po wpo 4443
This theorem is referenced by:  nfso  4451
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ral 2655  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-br 4155  df-po 4445
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