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Theorem ordeq 4294
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 4032 . . 3 (𝐴 = 𝐵 → (Tr 𝐴 ↔ Tr 𝐵))
2 raleq 2626 . . 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 4289 . 2 (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥𝐴 Tr 𝑥))
5 dford3 4289 . 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 1331  wral 2416  Tr wtr 4026  Ord word 4284
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-in 3077  df-ss 3084  df-uni 3737  df-tr 4027  df-iord 4288
This theorem is referenced by:  elong  4295  limeq  4299  ordelord  4303  ordtriexmidlem  4435  2ordpr  4439  issmo  6185  issmo2  6186  smoeq  6187  smores  6189  smores2  6191  smodm2  6192  smoiso  6199  tfrlem8  6215  tfri1dALT  6248
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