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Theorem nfso 5598
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 5592 . 2 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑎𝐴𝑏𝐴 (𝑎𝑅𝑏𝑎 = 𝑏𝑏𝑅𝑎)))
2 nfpo.r . . . 4 𝑥𝑅
3 nfpo.a . . . 4 𝑥𝐴
42, 3nfpo 5597 . . 3 𝑥 𝑅 Po 𝐴
5 nfcv 2904 . . . . . . 7 𝑥𝑎
6 nfcv 2904 . . . . . . 7 𝑥𝑏
75, 2, 6nfbr 5189 . . . . . 6 𝑥 𝑎𝑅𝑏
8 nfv 1913 . . . . . 6 𝑥 𝑎 = 𝑏
96, 2, 5nfbr 5189 . . . . . 6 𝑥 𝑏𝑅𝑎
107, 8, 9nf3or 1904 . . . . 5 𝑥(𝑎𝑅𝑏𝑎 = 𝑏𝑏𝑅𝑎)
113, 10nfralw 3310 . . . 4 𝑥𝑏𝐴 (𝑎𝑅𝑏𝑎 = 𝑏𝑏𝑅𝑎)
123, 11nfralw 3310 . . 3 𝑥𝑎𝐴𝑏𝐴 (𝑎𝑅𝑏𝑎 = 𝑏𝑏𝑅𝑎)
134, 12nfan 1898 . 2 𝑥(𝑅 Po 𝐴 ∧ ∀𝑎𝐴𝑏𝐴 (𝑎𝑅𝑏𝑎 = 𝑏𝑏𝑅𝑎))
141, 13nfxfr 1852 1 𝑥 𝑅 Or 𝐴
Colors of variables: wff setvar class
Syntax hints:  wa 395  w3o 1085  wnf 1782  wnfc 2889  wral 3060   class class class wbr 5142   Po wpo 5589   Or wor 5590
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-po 5591  df-so 5592
This theorem is referenced by:  nfwe  5659  weiunso  36468
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