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Theorem nfpo 4066
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 4061 . 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 2194 . . . . . . . 8  |-  F/_ x
a
4 nfpo.r . . . . . . . 8  |-  F/_ x R
53, 4, 3nfbr 3836 . . . . . . 7  |-  F/ x  a R a
65nfn 1564 . . . . . 6  |-  F/ x  -.  a R a
7 nfcv 2194 . . . . . . . . 9  |-  F/_ x
b
83, 4, 7nfbr 3836 . . . . . . . 8  |-  F/ x  a R b
9 nfcv 2194 . . . . . . . . 9  |-  F/_ x
c
107, 4, 9nfbr 3836 . . . . . . . 8  |-  F/ x  b R c
118, 10nfan 1473 . . . . . . 7  |-  F/ x
( a R b  /\  b R c )
123, 4, 9nfbr 3836 . . . . . . 7  |-  F/ x  a R c
1311, 12nfim 1480 . . . . . 6  |-  F/ x
( ( a R b  /\  b R c )  ->  a R c )
146, 13nfan 1473 . . . . 5  |-  F/ x
( -.  a R a  /\  ( ( a R b  /\  b R c )  -> 
a R c ) )
152, 14nfralxy 2377 . . . 4  |-  F/ x A. c  e.  A  ( -.  a R
a  /\  ( (
a R b  /\  b R c )  -> 
a R c ) )
162, 15nfralxy 2377 . . 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 2377 . 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 1379 1  |-  F/ x  R  Po  A
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 101   F/wnf 1365   F/_wnfc 2181   A.wral 2323   class class class wbr 3792    Po wpo 4059
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-v 2576  df-un 2950  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-po 4061
This theorem is referenced by:  nfso  4067
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