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Theorem nfpo 4029
Description: Bound-variable hypothesis builder for partial orders. (Contributed by Stefan O'Rear, 20-Jan-2015.)
Hypotheses
Ref Expression
nfpo.r  F/_ R
nfpo.a  F/_
Assertion
Ref Expression
nfpo  F/  R  Po

Proof of Theorem nfpo
Dummy variables  a  b  c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-po 4024 . 2  R  Po  a  b  c  a R a  a R b  b R c  a R c
2 nfpo.a . . 3  F/_
3 nfcv 2175 . . . . . . . 8  F/_ a
4 nfpo.r . . . . . . . 8  F/_ R
53, 4, 3nfbr 3799 . . . . . . 7  F/  a R a
65nfn 1545 . . . . . 6  F/  a R a
7 nfcv 2175 . . . . . . . . 9  F/_ b
83, 4, 7nfbr 3799 . . . . . . . 8  F/  a R b
9 nfcv 2175 . . . . . . . . 9  F/_ c
107, 4, 9nfbr 3799 . . . . . . . 8  F/  b R c
118, 10nfan 1454 . . . . . . 7  F/ a R b  b R c
123, 4, 9nfbr 3799 . . . . . . 7  F/  a R c
1311, 12nfim 1461 . . . . . 6  F/ a R b  b R c  a R c
146, 13nfan 1454 . . . . 5  F/  a R a  a R b  b R c  a R c
152, 14nfralxy 2354 . . . 4  F/ c  a R a  a R b  b R c  a R c
162, 15nfralxy 2354 . . 3  F/ b  c  a R a  a R b  b R c  a R c
172, 16nfralxy 2354 . 2  F/ a  b  c  a R a  a R b  b R c  a R c
181, 17nfxfr 1360 1  F/  R  Po
Colors of variables: wff set class
Syntax hints:   wn 3   wi 4   wa 97   F/wnf 1346   F/_wnfc 2162  wral 2300   class class class wbr 3755    Po wpo 4022
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-po 4024
This theorem is referenced by:  nfso  4030
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