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Theorem ordeq 4137
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 3888 . . 3 (𝐴 = 𝐵 → (Tr 𝐴 ↔ Tr 𝐵))
2 raleq 2522 . . 3 (𝐴 = 𝐵 → (∀𝑥𝐴 Tr 𝑥 ↔ ∀𝑥𝐵 Tr 𝑥))
31, 2anbi12d 450 . 2 (𝐴 = 𝐵 → ((Tr 𝐴 ∧ ∀𝑥𝐴 Tr 𝑥) ↔ (Tr 𝐵 ∧ ∀𝑥𝐵 Tr 𝑥)))
4 dford3 4132 . 2 (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥𝐴 Tr 𝑥))
5 dford3 4132 . 2 (Ord 𝐵 ↔ (Tr 𝐵 ∧ ∀𝑥𝐵 Tr 𝑥))
63, 4, 53bitr4g 216 1 (𝐴 = 𝐵 → (Ord 𝐴 ↔ Ord 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wral 2323  Tr wtr 3882  Ord word 4127
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-in 2952  df-ss 2959  df-uni 3609  df-tr 3883  df-iord 4131
This theorem is referenced by:  elong  4138  limeq  4142  ordelord  4146  ordtriexmidlem  4273  2ordpr  4277  issmo  5934  issmo2  5935  smoeq  5936  smores  5938  smores2  5940  smodm2  5941  smoiso  5948  tfrlem8  5965
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