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Theorem nfpo 4223
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 4218 . 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 2281 . . . . . . . 8  |-  F/_ x
a
4 nfpo.r . . . . . . . 8  |-  F/_ x R
53, 4, 3nfbr 3974 . . . . . . 7  |-  F/ x  a R a
65nfn 1636 . . . . . 6  |-  F/ x  -.  a R a
7 nfcv 2281 . . . . . . . . 9  |-  F/_ x
b
83, 4, 7nfbr 3974 . . . . . . . 8  |-  F/ x  a R b
9 nfcv 2281 . . . . . . . . 9  |-  F/_ x
c
107, 4, 9nfbr 3974 . . . . . . . 8  |-  F/ x  b R c
118, 10nfan 1544 . . . . . . 7  |-  F/ x
( a R b  /\  b R c )
123, 4, 9nfbr 3974 . . . . . . 7  |-  F/ x  a R c
1311, 12nfim 1551 . . . . . 6  |-  F/ x
( ( a R b  /\  b R c )  ->  a R c )
146, 13nfan 1544 . . . . 5  |-  F/ x
( -.  a R a  /\  ( ( a R b  /\  b R c )  -> 
a R c ) )
152, 14nfralxy 2471 . . . 4  |-  F/ x A. c  e.  A  ( -.  a R
a  /\  ( (
a R b  /\  b R c )  -> 
a R c ) )
162, 15nfralxy 2471 . . 3  |-  F/ x A. b  e.  A  A. c  e.  A  ( -.  a R
a  /\  ( (
a R b  /\  b R c )  -> 
a R c ) )
172, 16nfralxy 2471 . 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 1450 1  |-  F/ x  R  Po  A
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103   F/wnf 1436   F/_wnfc 2268   A.wral 2416   class class class wbr 3929    Po wpo 4216
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-po 4218
This theorem is referenced by:  nfso  4224
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