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Theorem nfso 5473
Description: Bound-variable hypothesis builder for total orders. (Contributed by Stefan O'Rear, 20-Jan-2015.)
Hypotheses
Ref Expression
nfpo.r 𝑥𝑅
nfpo.a 𝑥𝐴
Assertion
Ref Expression
nfso 𝑥 𝑅 Or 𝐴

Proof of Theorem nfso
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-so 5468 . 2 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑎𝐴𝑏𝐴 (𝑎𝑅𝑏𝑎 = 𝑏𝑏𝑅𝑎)))
2 nfpo.r . . . 4 𝑥𝑅
3 nfpo.a . . . 4 𝑥𝐴
42, 3nfpo 5472 . . 3 𝑥 𝑅 Po 𝐴
5 nfcv 2974 . . . . . . 7 𝑥𝑎
6 nfcv 2974 . . . . . . 7 𝑥𝑏
75, 2, 6nfbr 5104 . . . . . 6 𝑥 𝑎𝑅𝑏
8 nfv 1906 . . . . . 6 𝑥 𝑎 = 𝑏
96, 2, 5nfbr 5104 . . . . . 6 𝑥 𝑏𝑅𝑎
107, 8, 9nf3or 1897 . . . . 5 𝑥(𝑎𝑅𝑏𝑎 = 𝑏𝑏𝑅𝑎)
113, 10nfralw 3222 . . . 4 𝑥𝑏𝐴 (𝑎𝑅𝑏𝑎 = 𝑏𝑏𝑅𝑎)
123, 11nfralw 3222 . . 3 𝑥𝑎𝐴𝑏𝐴 (𝑎𝑅𝑏𝑎 = 𝑏𝑏𝑅𝑎)
134, 12nfan 1891 . 2 𝑥(𝑅 Po 𝐴 ∧ ∀𝑎𝐴𝑏𝐴 (𝑎𝑅𝑏𝑎 = 𝑏𝑏𝑅𝑎))
141, 13nfxfr 1844 1 𝑥 𝑅 Or 𝐴
Colors of variables: wff setvar class
Syntax hints:  wa 396  w3o 1078  wnf 1775  wnfc 2958  wral 3135   class class class wbr 5057   Po wpo 5465   Or wor 5466
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-3or 1080  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  df-so 5468
This theorem is referenced by:  nfwe  5524
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