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Theorem ordeq 4208
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 3948 . . 3 (𝐴 = 𝐵 → (Tr 𝐴 ↔ Tr 𝐵))
2 raleq 2563 . . 3 (𝐴 = 𝐵 → (∀𝑥𝐴 Tr 𝑥 ↔ ∀𝑥𝐵 Tr 𝑥))
31, 2anbi12d 458 . 2 (𝐴 = 𝐵 → ((Tr 𝐴 ∧ ∀𝑥𝐴 Tr 𝑥) ↔ (Tr 𝐵 ∧ ∀𝑥𝐵 Tr 𝑥)))
4 dford3 4203 . 2 (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥𝐴 Tr 𝑥))
5 dford3 4203 . 2 (Ord 𝐵 ↔ (Tr 𝐵 ∧ ∀𝑥𝐵 Tr 𝑥))
63, 4, 53bitr4g 222 1 (𝐴 = 𝐵 → (Ord 𝐴 ↔ Ord 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1290  wral 2360  Tr wtr 3942  Ord word 4198
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-in 3006  df-ss 3013  df-uni 3660  df-tr 3943  df-iord 4202
This theorem is referenced by:  elong  4209  limeq  4213  ordelord  4217  ordtriexmidlem  4349  2ordpr  4353  issmo  6067  issmo2  6068  smoeq  6069  smores  6071  smores2  6073  smodm2  6074  smoiso  6081  tfrlem8  6097  tfri1dALT  6130
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