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Theorem nfpo 5000
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 4995 . 2 (𝑅 Po 𝐴 ↔ ∀𝑎𝐴𝑏𝐴𝑐𝐴𝑎𝑅𝑎 ∧ ((𝑎𝑅𝑏𝑏𝑅𝑐) → 𝑎𝑅𝑐)))
2 nfpo.a . . 3 𝑥𝐴
3 nfcv 2761 . . . . . . . 8 𝑥𝑎
4 nfpo.r . . . . . . . 8 𝑥𝑅
53, 4, 3nfbr 4659 . . . . . . 7 𝑥 𝑎𝑅𝑎
65nfn 1781 . . . . . 6 𝑥 ¬ 𝑎𝑅𝑎
7 nfcv 2761 . . . . . . . . 9 𝑥𝑏
83, 4, 7nfbr 4659 . . . . . . . 8 𝑥 𝑎𝑅𝑏
9 nfcv 2761 . . . . . . . . 9 𝑥𝑐
107, 4, 9nfbr 4659 . . . . . . . 8 𝑥 𝑏𝑅𝑐
118, 10nfan 1825 . . . . . . 7 𝑥(𝑎𝑅𝑏𝑏𝑅𝑐)
123, 4, 9nfbr 4659 . . . . . . 7 𝑥 𝑎𝑅𝑐
1311, 12nfim 1822 . . . . . 6 𝑥((𝑎𝑅𝑏𝑏𝑅𝑐) → 𝑎𝑅𝑐)
146, 13nfan 1825 . . . . 5 𝑥𝑎𝑅𝑎 ∧ ((𝑎𝑅𝑏𝑏𝑅𝑐) → 𝑎𝑅𝑐))
152, 14nfral 2940 . . . 4 𝑥𝑐𝐴𝑎𝑅𝑎 ∧ ((𝑎𝑅𝑏𝑏𝑅𝑐) → 𝑎𝑅𝑐))
162, 15nfral 2940 . . 3 𝑥𝑏𝐴𝑐𝐴𝑎𝑅𝑎 ∧ ((𝑎𝑅𝑏𝑏𝑅𝑐) → 𝑎𝑅𝑐))
172, 16nfral 2940 . 2 𝑥𝑎𝐴𝑏𝐴𝑐𝐴𝑎𝑅𝑎 ∧ ((𝑎𝑅𝑏𝑏𝑅𝑐) → 𝑎𝑅𝑐))
181, 17nfxfr 1776 1 𝑥 𝑅 Po 𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  wnf 1705  wnfc 2748  wral 2907   class class class wbr 4613   Po wpo 4993
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-po 4995
This theorem is referenced by:  nfso  5001
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