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Theorem nfpo 5447
 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 5442 . 2 (𝑅 Po 𝐴 ↔ ∀𝑎𝐴𝑏𝐴𝑐𝐴𝑎𝑅𝑎 ∧ ((𝑎𝑅𝑏𝑏𝑅𝑐) → 𝑎𝑅𝑐)))
2 nfpo.a . . 3 𝑥𝐴
3 nfcv 2955 . . . . . . . 8 𝑥𝑎
4 nfpo.r . . . . . . . 8 𝑥𝑅
53, 4, 3nfbr 5081 . . . . . . 7 𝑥 𝑎𝑅𝑎
65nfn 1858 . . . . . 6 𝑥 ¬ 𝑎𝑅𝑎
7 nfcv 2955 . . . . . . . . 9 𝑥𝑏
83, 4, 7nfbr 5081 . . . . . . . 8 𝑥 𝑎𝑅𝑏
9 nfcv 2955 . . . . . . . . 9 𝑥𝑐
107, 4, 9nfbr 5081 . . . . . . . 8 𝑥 𝑏𝑅𝑐
118, 10nfan 1900 . . . . . . 7 𝑥(𝑎𝑅𝑏𝑏𝑅𝑐)
123, 4, 9nfbr 5081 . . . . . . 7 𝑥 𝑎𝑅𝑐
1311, 12nfim 1897 . . . . . 6 𝑥((𝑎𝑅𝑏𝑏𝑅𝑐) → 𝑎𝑅𝑐)
146, 13nfan 1900 . . . . 5 𝑥𝑎𝑅𝑎 ∧ ((𝑎𝑅𝑏𝑏𝑅𝑐) → 𝑎𝑅𝑐))
152, 14nfralw 3189 . . . 4 𝑥𝑐𝐴𝑎𝑅𝑎 ∧ ((𝑎𝑅𝑏𝑏𝑅𝑐) → 𝑎𝑅𝑐))
162, 15nfralw 3189 . . 3 𝑥𝑏𝐴𝑐𝐴𝑎𝑅𝑎 ∧ ((𝑎𝑅𝑏𝑏𝑅𝑐) → 𝑎𝑅𝑐))
172, 16nfralw 3189 . 2 𝑥𝑎𝐴𝑏𝐴𝑐𝐴𝑎𝑅𝑎 ∧ ((𝑎𝑅𝑏𝑏𝑅𝑐) → 𝑎𝑅𝑐))
181, 17nfxfr 1854 1 𝑥 𝑅 Po 𝐴
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399  Ⅎwnf 1785  Ⅎwnfc 2936  ∀wral 3106   class class class wbr 5034   Po wpo 5440 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-v 3444  df-dif 3886  df-un 3888  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5035  df-po 5442 This theorem is referenced by:  nfso  5448
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