ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ordeq GIF version

Theorem ordeq 4368
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 4104 . . 3 (𝐴 = 𝐵 → (Tr 𝐴 ↔ Tr 𝐵))
2 raleq 2672 . . 3 (𝐴 = 𝐵 → (∀𝑥𝐴 Tr 𝑥 ↔ ∀𝑥𝐵 Tr 𝑥))
31, 2anbi12d 473 . 2 (𝐴 = 𝐵 → ((Tr 𝐴 ∧ ∀𝑥𝐴 Tr 𝑥) ↔ (Tr 𝐵 ∧ ∀𝑥𝐵 Tr 𝑥)))
4 dford3 4363 . 2 (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥𝐴 Tr 𝑥))
5 dford3 4363 . 2 (Ord 𝐵 ↔ (Tr 𝐵 ∧ ∀𝑥𝐵 Tr 𝑥))
63, 4, 53bitr4g 223 1 (𝐴 = 𝐵 → (Ord 𝐴 ↔ Ord 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wral 2455  Tr wtr 4098  Ord word 4358
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-in 3135  df-ss 3142  df-uni 3808  df-tr 4099  df-iord 4362
This theorem is referenced by:  elong  4369  limeq  4373  ordelord  4377  ordtriexmidlem  4514  2ordpr  4519  issmo  6282  issmo2  6283  smoeq  6284  smores  6286  smores2  6288  smodm2  6289  smoiso  6296  tfrlem8  6312  tfri1dALT  6345
  Copyright terms: Public domain W3C validator