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Theorem treq 3934
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 3657 . . . 4 (𝐴 = 𝐵 𝐴 = 𝐵)
21sseq1d 3051 . . 3 (𝐴 = 𝐵 → ( 𝐴𝐴 𝐵𝐴))
3 sseq2 3046 . . 3 (𝐴 = 𝐵 → ( 𝐵𝐴 𝐵𝐵))
42, 3bitrd 186 . 2 (𝐴 = 𝐵 → ( 𝐴𝐴 𝐵𝐵))
5 df-tr 3929 . 2 (Tr 𝐴 𝐴𝐴)
6 df-tr 3929 . 2 (Tr 𝐵 𝐵𝐵)
74, 5, 63bitr4g 221 1 (𝐴 = 𝐵 → (Tr 𝐴 ↔ Tr 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103   = wceq 1289  wss 2997   cuni 3648  Tr wtr 3928
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-in 3003  df-ss 3010  df-uni 3649  df-tr 3929
This theorem is referenced by:  truni  3942  ordeq  4190  ordsucim  4307  ordom  4411
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