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Theorem nfso 4029
 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 4024 . 2 (𝑅 Or A ↔ (𝑅 Po A 𝑎 A 𝑏 A 𝑐 A (𝑎𝑅𝑏 → (𝑎𝑅𝑐 𝑐𝑅𝑏))))
2 nfpo.r . . . 4 x𝑅
3 nfpo.a . . . 4 xA
42, 3nfpo 4028 . . 3 x 𝑅 Po A
5 nfcv 2175 . . . . . . . 8 x𝑎
6 nfcv 2175 . . . . . . . 8 x𝑏
75, 2, 6nfbr 3798 . . . . . . 7 x 𝑎𝑅𝑏
8 nfcv 2175 . . . . . . . . 9 x𝑐
95, 2, 8nfbr 3798 . . . . . . . 8 x 𝑎𝑅𝑐
108, 2, 6nfbr 3798 . . . . . . . 8 x 𝑐𝑅𝑏
119, 10nfor 1463 . . . . . . 7 x(𝑎𝑅𝑐 𝑐𝑅𝑏)
127, 11nfim 1461 . . . . . 6 x(𝑎𝑅𝑏 → (𝑎𝑅𝑐 𝑐𝑅𝑏))
133, 12nfralxy 2354 . . . . 5 x𝑐 A (𝑎𝑅𝑏 → (𝑎𝑅𝑐 𝑐𝑅𝑏))
143, 13nfralxy 2354 . . . 4 x𝑏 A 𝑐 A (𝑎𝑅𝑏 → (𝑎𝑅𝑐 𝑐𝑅𝑏))
153, 14nfralxy 2354 . . 3 x𝑎 A 𝑏 A 𝑐 A (𝑎𝑅𝑏 → (𝑎𝑅𝑐 𝑐𝑅𝑏))
164, 15nfan 1454 . 2 x(𝑅 Po A 𝑎 A 𝑏 A 𝑐 A (𝑎𝑅𝑏 → (𝑎𝑅𝑐 𝑐𝑅𝑏)))
171, 16nfxfr 1360 1 x 𝑅 Or A
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ∨ wo 628  Ⅎwnf 1346  Ⅎwnfc 2162  ∀wral 2300   class class class wbr 3754   Po wpo 4021   Or wor 4022 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-un 2916  df-sn 3372  df-pr 3373  df-op 3375  df-br 3755  df-po 4023  df-iso 4024 This theorem is referenced by: (None)
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