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Theorem nfpo 5472
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 5467 . 2 (𝑅 Po 𝐴 ↔ ∀𝑎𝐴𝑏𝐴𝑐𝐴𝑎𝑅𝑎 ∧ ((𝑎𝑅𝑏𝑏𝑅𝑐) → 𝑎𝑅𝑐)))
2 nfpo.a . . 3 𝑥𝐴
3 nfcv 2974 . . . . . . . 8 𝑥𝑎
4 nfpo.r . . . . . . . 8 𝑥𝑅
53, 4, 3nfbr 5104 . . . . . . 7 𝑥 𝑎𝑅𝑎
65nfn 1848 . . . . . 6 𝑥 ¬ 𝑎𝑅𝑎
7 nfcv 2974 . . . . . . . . 9 𝑥𝑏
83, 4, 7nfbr 5104 . . . . . . . 8 𝑥 𝑎𝑅𝑏
9 nfcv 2974 . . . . . . . . 9 𝑥𝑐
107, 4, 9nfbr 5104 . . . . . . . 8 𝑥 𝑏𝑅𝑐
118, 10nfan 1891 . . . . . . 7 𝑥(𝑎𝑅𝑏𝑏𝑅𝑐)
123, 4, 9nfbr 5104 . . . . . . 7 𝑥 𝑎𝑅𝑐
1311, 12nfim 1888 . . . . . 6 𝑥((𝑎𝑅𝑏𝑏𝑅𝑐) → 𝑎𝑅𝑐)
146, 13nfan 1891 . . . . 5 𝑥𝑎𝑅𝑎 ∧ ((𝑎𝑅𝑏𝑏𝑅𝑐) → 𝑎𝑅𝑐))
152, 14nfralw 3222 . . . 4 𝑥𝑐𝐴𝑎𝑅𝑎 ∧ ((𝑎𝑅𝑏𝑏𝑅𝑐) → 𝑎𝑅𝑐))
162, 15nfralw 3222 . . 3 𝑥𝑏𝐴𝑐𝐴𝑎𝑅𝑎 ∧ ((𝑎𝑅𝑏𝑏𝑅𝑐) → 𝑎𝑅𝑐))
172, 16nfralw 3222 . 2 𝑥𝑎𝐴𝑏𝐴𝑐𝐴𝑎𝑅𝑎 ∧ ((𝑎𝑅𝑏𝑏𝑅𝑐) → 𝑎𝑅𝑐))
181, 17nfxfr 1844 1 𝑥 𝑅 Po 𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wnf 1775  wnfc 2958  wral 3135   class class class wbr 5057   Po wpo 5465
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-po 5467
This theorem is referenced by:  nfso  5473
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