ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nfso GIF version

Theorem nfso 4105
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 4100 . 2 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑎𝐴𝑏𝐴𝑐𝐴 (𝑎𝑅𝑏 → (𝑎𝑅𝑐𝑐𝑅𝑏))))
2 nfpo.r . . . 4 𝑥𝑅
3 nfpo.a . . . 4 𝑥𝐴
42, 3nfpo 4104 . . 3 𝑥 𝑅 Po 𝐴
5 nfcv 2225 . . . . . . . 8 𝑥𝑎
6 nfcv 2225 . . . . . . . 8 𝑥𝑏
75, 2, 6nfbr 3866 . . . . . . 7 𝑥 𝑎𝑅𝑏
8 nfcv 2225 . . . . . . . . 9 𝑥𝑐
95, 2, 8nfbr 3866 . . . . . . . 8 𝑥 𝑎𝑅𝑐
108, 2, 6nfbr 3866 . . . . . . . 8 𝑥 𝑐𝑅𝑏
119, 10nfor 1509 . . . . . . 7 𝑥(𝑎𝑅𝑐𝑐𝑅𝑏)
127, 11nfim 1507 . . . . . 6 𝑥(𝑎𝑅𝑏 → (𝑎𝑅𝑐𝑐𝑅𝑏))
133, 12nfralxy 2410 . . . . 5 𝑥𝑐𝐴 (𝑎𝑅𝑏 → (𝑎𝑅𝑐𝑐𝑅𝑏))
143, 13nfralxy 2410 . . . 4 𝑥𝑏𝐴𝑐𝐴 (𝑎𝑅𝑏 → (𝑎𝑅𝑐𝑐𝑅𝑏))
153, 14nfralxy 2410 . . 3 𝑥𝑎𝐴𝑏𝐴𝑐𝐴 (𝑎𝑅𝑏 → (𝑎𝑅𝑐𝑐𝑅𝑏))
164, 15nfan 1500 . 2 𝑥(𝑅 Po 𝐴 ∧ ∀𝑎𝐴𝑏𝐴𝑐𝐴 (𝑎𝑅𝑏 → (𝑎𝑅𝑐𝑐𝑅𝑏)))
171, 16nfxfr 1406 1 𝑥 𝑅 Or 𝐴
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wo 662  wnf 1392  wnfc 2212  wral 2355   class class class wbr 3822   Po wpo 4097   Or wor 4098
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-v 2617  df-un 2992  df-sn 3437  df-pr 3438  df-op 3440  df-br 3823  df-po 4099  df-iso 4100
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator