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Theorem ordeq 4264
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 4002 . . 3 (𝐴 = 𝐵 → (Tr 𝐴 ↔ Tr 𝐵))
2 raleq 2603 . . 3 (𝐴 = 𝐵 → (∀𝑥𝐴 Tr 𝑥 ↔ ∀𝑥𝐵 Tr 𝑥))
31, 2anbi12d 464 . 2 (𝐴 = 𝐵 → ((Tr 𝐴 ∧ ∀𝑥𝐴 Tr 𝑥) ↔ (Tr 𝐵 ∧ ∀𝑥𝐵 Tr 𝑥)))
4 dford3 4259 . 2 (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥𝐴 Tr 𝑥))
5 dford3 4259 . 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 1316  wral 2393  Tr wtr 3996  Ord word 4254
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-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-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-in 3047  df-ss 3054  df-uni 3707  df-tr 3997  df-iord 4258
This theorem is referenced by:  elong  4265  limeq  4269  ordelord  4273  ordtriexmidlem  4405  2ordpr  4409  issmo  6153  issmo2  6154  smoeq  6155  smores  6157  smores2  6159  smodm2  6160  smoiso  6167  tfrlem8  6183  tfri1dALT  6216
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