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Theorem ordeq 4075
Description: Equality theorem for the ordinal predicate. (Contributed by NM, 17-Sep-1993.)
Assertion
Ref Expression
ordeq (A = B → (Ord A ↔ Ord B))

Proof of Theorem ordeq
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 treq 3851 . . 3 (A = B → (Tr A ↔ Tr B))
2 raleq 2499 . . 3 (A = B → (x A Tr xx B Tr x))
31, 2anbi12d 442 . 2 (A = B → ((Tr A x A Tr x) ↔ (Tr B x B Tr x)))
4 dford3 4070 . 2 (Ord A ↔ (Tr A x A Tr x))
5 dford3 4070 . 2 (Ord B ↔ (Tr B x B Tr x))
63, 4, 53bitr4g 212 1 (A = B → (Ord A ↔ Ord B))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242  wral 2300  Tr wtr 3845  Ord word 4065
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-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-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-in 2918  df-ss 2925  df-uni 3572  df-tr 3846  df-iord 4069
This theorem is referenced by:  elong  4076  limeq  4080  ordelord  4084  ordtriexmidlem  4208  onsucelsucexmidlem  4214  issmo  5844  issmo2  5845  smoeq  5846  smores  5848  smores2  5850  smodm2  5851  smoiso  5858  tfrlem8  5875
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