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Theorem nfpo 4028
Description: Bound-variable hypothesis builder for partial orders. (Contributed by Stefan O'Rear, 20-Jan-2015.)
Hypotheses
Ref Expression
nfpo.r x𝑅
nfpo.a xA
Assertion
Ref Expression
nfpo 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 4023 . 2 (𝑅 Po A𝑎 A 𝑏 A 𝑐 A𝑎𝑅𝑎 ((𝑎𝑅𝑏 𝑏𝑅𝑐) → 𝑎𝑅𝑐)))
2 nfpo.a . . 3 xA
3 nfcv 2175 . . . . . . . 8 x𝑎
4 nfpo.r . . . . . . . 8 x𝑅
53, 4, 3nfbr 3798 . . . . . . 7 x 𝑎𝑅𝑎
65nfn 1545 . . . . . 6 x ¬ 𝑎𝑅𝑎
7 nfcv 2175 . . . . . . . . 9 x𝑏
83, 4, 7nfbr 3798 . . . . . . . 8 x 𝑎𝑅𝑏
9 nfcv 2175 . . . . . . . . 9 x𝑐
107, 4, 9nfbr 3798 . . . . . . . 8 x 𝑏𝑅𝑐
118, 10nfan 1454 . . . . . . 7 x(𝑎𝑅𝑏 𝑏𝑅𝑐)
123, 4, 9nfbr 3798 . . . . . . 7 x 𝑎𝑅𝑐
1311, 12nfim 1461 . . . . . 6 x((𝑎𝑅𝑏 𝑏𝑅𝑐) → 𝑎𝑅𝑐)
146, 13nfan 1454 . . . . 5 x𝑎𝑅𝑎 ((𝑎𝑅𝑏 𝑏𝑅𝑐) → 𝑎𝑅𝑐))
152, 14nfralxy 2354 . . . 4 x𝑐 A𝑎𝑅𝑎 ((𝑎𝑅𝑏 𝑏𝑅𝑐) → 𝑎𝑅𝑐))
162, 15nfralxy 2354 . . 3 x𝑏 A 𝑐 A𝑎𝑅𝑎 ((𝑎𝑅𝑏 𝑏𝑅𝑐) → 𝑎𝑅𝑐))
172, 16nfralxy 2354 . 2 x𝑎 A 𝑏 A 𝑐 A𝑎𝑅𝑎 ((𝑎𝑅𝑏 𝑏𝑅𝑐) → 𝑎𝑅𝑐))
181, 17nfxfr 1360 1 x 𝑅 Po A
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wnf 1346  wnfc 2162  wral 2300   class class class wbr 3754   Po wpo 4021
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-bnd 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 3372  df-pr 3373  df-op 3375  df-br 3755  df-po 4023
This theorem is referenced by:  nfso  4029
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