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Theorem treq 3851
Description: Equality theorem for the transitive class predicate. (Contributed by NM, 17-Sep-1993.)
Assertion
Ref Expression
treq (A = B → (Tr A ↔ Tr B))

Proof of Theorem treq
StepHypRef Expression
1 unieq 3580 . . . 4 (A = B A = B)
21sseq1d 2966 . . 3 (A = B → ( AA BA))
3 sseq2 2961 . . 3 (A = B → ( BA BB))
42, 3bitrd 177 . 2 (A = B → ( AA BB))
5 df-tr 3846 . 2 (Tr A AA)
6 df-tr 3846 . 2 (Tr B BB)
74, 5, 63bitr4g 212 1 (A = B → (Tr A ↔ Tr B))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1242  wss 2911   cuni 3571  Tr wtr 3845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-in 2918  df-ss 2925  df-uni 3572  df-tr 3846
This theorem is referenced by:  truni  3859  ordeq  4075  ordsucim  4192  ordom  4272
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