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Theorem nfso 4194
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 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-iso 4189 . 2 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑎𝐴𝑏𝐴𝑐𝐴 (𝑎𝑅𝑏 → (𝑎𝑅𝑐𝑐𝑅𝑏))))
2 nfpo.r . . . 4 𝑥𝑅
3 nfpo.a . . . 4 𝑥𝐴
42, 3nfpo 4193 . . 3 𝑥 𝑅 Po 𝐴
5 nfcv 2258 . . . . . . . 8 𝑥𝑎
6 nfcv 2258 . . . . . . . 8 𝑥𝑏
75, 2, 6nfbr 3944 . . . . . . 7 𝑥 𝑎𝑅𝑏
8 nfcv 2258 . . . . . . . . 9 𝑥𝑐
95, 2, 8nfbr 3944 . . . . . . . 8 𝑥 𝑎𝑅𝑐
108, 2, 6nfbr 3944 . . . . . . . 8 𝑥 𝑐𝑅𝑏
119, 10nfor 1538 . . . . . . 7 𝑥(𝑎𝑅𝑐𝑐𝑅𝑏)
127, 11nfim 1536 . . . . . 6 𝑥(𝑎𝑅𝑏 → (𝑎𝑅𝑐𝑐𝑅𝑏))
133, 12nfralxy 2448 . . . . 5 𝑥𝑐𝐴 (𝑎𝑅𝑏 → (𝑎𝑅𝑐𝑐𝑅𝑏))
143, 13nfralxy 2448 . . . 4 𝑥𝑏𝐴𝑐𝐴 (𝑎𝑅𝑏 → (𝑎𝑅𝑐𝑐𝑅𝑏))
153, 14nfralxy 2448 . . 3 𝑥𝑎𝐴𝑏𝐴𝑐𝐴 (𝑎𝑅𝑏 → (𝑎𝑅𝑐𝑐𝑅𝑏))
164, 15nfan 1529 . 2 𝑥(𝑅 Po 𝐴 ∧ ∀𝑎𝐴𝑏𝐴𝑐𝐴 (𝑎𝑅𝑏 → (𝑎𝑅𝑐𝑐𝑅𝑏)))
171, 16nfxfr 1435 1 𝑥 𝑅 Or 𝐴
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wo 682  wnf 1421  wnfc 2245  wral 2393   class class class wbr 3899   Po wpo 4186   Or wor 4187
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-v 2662  df-un 3045  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-po 4188  df-iso 4189
This theorem is referenced by: (None)
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