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Theorem nfso 5001
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 4996 . 2 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑎𝐴𝑏𝐴 (𝑎𝑅𝑏𝑎 = 𝑏𝑏𝑅𝑎)))
2 nfpo.r . . . 4 𝑥𝑅
3 nfpo.a . . . 4 𝑥𝐴
42, 3nfpo 5000 . . 3 𝑥 𝑅 Po 𝐴
5 nfcv 2761 . . . . . . 7 𝑥𝑎
6 nfcv 2761 . . . . . . 7 𝑥𝑏
75, 2, 6nfbr 4659 . . . . . 6 𝑥 𝑎𝑅𝑏
8 nfv 1840 . . . . . 6 𝑥 𝑎 = 𝑏
96, 2, 5nfbr 4659 . . . . . 6 𝑥 𝑏𝑅𝑎
107, 8, 9nf3or 1832 . . . . 5 𝑥(𝑎𝑅𝑏𝑎 = 𝑏𝑏𝑅𝑎)
113, 10nfral 2940 . . . 4 𝑥𝑏𝐴 (𝑎𝑅𝑏𝑎 = 𝑏𝑏𝑅𝑎)
123, 11nfral 2940 . . 3 𝑥𝑎𝐴𝑏𝐴 (𝑎𝑅𝑏𝑎 = 𝑏𝑏𝑅𝑎)
134, 12nfan 1825 . 2 𝑥(𝑅 Po 𝐴 ∧ ∀𝑎𝐴𝑏𝐴 (𝑎𝑅𝑏𝑎 = 𝑏𝑏𝑅𝑎))
141, 13nfxfr 1776 1 𝑥 𝑅 Or 𝐴
Colors of variables: wff setvar class
Syntax hints:  wa 384  w3o 1035  wnf 1705  wnfc 2748  wral 2907   class class class wbr 4613   Po wpo 4993   Or wor 4994
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-3or 1037  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  df-so 4996
This theorem is referenced by:  nfwe  5050
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