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Theorem nfso 5551
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 5546 . 2 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑎𝐴𝑏𝐴 (𝑎𝑅𝑏𝑎 = 𝑏𝑏𝑅𝑎)))
2 nfpo.r . . . 4 𝑥𝑅
3 nfpo.a . . . 4 𝑥𝐴
42, 3nfpo 5550 . . 3 𝑥 𝑅 Po 𝐴
5 nfcv 2907 . . . . . . 7 𝑥𝑎
6 nfcv 2907 . . . . . . 7 𝑥𝑏
75, 2, 6nfbr 5152 . . . . . 6 𝑥 𝑎𝑅𝑏
8 nfv 1917 . . . . . 6 𝑥 𝑎 = 𝑏
96, 2, 5nfbr 5152 . . . . . 6 𝑥 𝑏𝑅𝑎
107, 8, 9nf3or 1908 . . . . 5 𝑥(𝑎𝑅𝑏𝑎 = 𝑏𝑏𝑅𝑎)
113, 10nfralw 3294 . . . 4 𝑥𝑏𝐴 (𝑎𝑅𝑏𝑎 = 𝑏𝑏𝑅𝑎)
123, 11nfralw 3294 . . 3 𝑥𝑎𝐴𝑏𝐴 (𝑎𝑅𝑏𝑎 = 𝑏𝑏𝑅𝑎)
134, 12nfan 1902 . 2 𝑥(𝑅 Po 𝐴 ∧ ∀𝑎𝐴𝑏𝐴 (𝑎𝑅𝑏𝑎 = 𝑏𝑏𝑅𝑎))
141, 13nfxfr 1855 1 𝑥 𝑅 Or 𝐴
Colors of variables: wff setvar class
Syntax hints:  wa 396  w3o 1086  wnf 1785  wnfc 2887  wral 3064   class class class wbr 5105   Po wpo 5543   Or wor 5544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ral 3065  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-br 5106  df-po 5545  df-so 5546
This theorem is referenced by:  nfwe  5609
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