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Theorem nfpo 4193
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 4188 . 2 (𝑅 Po 𝐴 ↔ ∀𝑎𝐴𝑏𝐴𝑐𝐴𝑎𝑅𝑎 ∧ ((𝑎𝑅𝑏𝑏𝑅𝑐) → 𝑎𝑅𝑐)))
2 nfpo.a . . 3 𝑥𝐴
3 nfcv 2258 . . . . . . . 8 𝑥𝑎
4 nfpo.r . . . . . . . 8 𝑥𝑅
53, 4, 3nfbr 3944 . . . . . . 7 𝑥 𝑎𝑅𝑎
65nfn 1621 . . . . . 6 𝑥 ¬ 𝑎𝑅𝑎
7 nfcv 2258 . . . . . . . . 9 𝑥𝑏
83, 4, 7nfbr 3944 . . . . . . . 8 𝑥 𝑎𝑅𝑏
9 nfcv 2258 . . . . . . . . 9 𝑥𝑐
107, 4, 9nfbr 3944 . . . . . . . 8 𝑥 𝑏𝑅𝑐
118, 10nfan 1529 . . . . . . 7 𝑥(𝑎𝑅𝑏𝑏𝑅𝑐)
123, 4, 9nfbr 3944 . . . . . . 7 𝑥 𝑎𝑅𝑐
1311, 12nfim 1536 . . . . . 6 𝑥((𝑎𝑅𝑏𝑏𝑅𝑐) → 𝑎𝑅𝑐)
146, 13nfan 1529 . . . . 5 𝑥𝑎𝑅𝑎 ∧ ((𝑎𝑅𝑏𝑏𝑅𝑐) → 𝑎𝑅𝑐))
152, 14nfralxy 2448 . . . 4 𝑥𝑐𝐴𝑎𝑅𝑎 ∧ ((𝑎𝑅𝑏𝑏𝑅𝑐) → 𝑎𝑅𝑐))
162, 15nfralxy 2448 . . 3 𝑥𝑏𝐴𝑐𝐴𝑎𝑅𝑎 ∧ ((𝑎𝑅𝑏𝑏𝑅𝑐) → 𝑎𝑅𝑐))
172, 16nfralxy 2448 . 2 𝑥𝑎𝐴𝑏𝐴𝑐𝐴𝑎𝑅𝑎 ∧ ((𝑎𝑅𝑏𝑏𝑅𝑐) → 𝑎𝑅𝑐))
181, 17nfxfr 1435 1 𝑥 𝑅 Po 𝐴
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wnf 1421  wnfc 2245  wral 2393   class class class wbr 3899   Po wpo 4186
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-v 2662  df-un 3045  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-po 4188
This theorem is referenced by:  nfso  4194
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