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Theorem treq 3888
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 3617 . . . 4 (𝐴 = 𝐵 𝐴 = 𝐵)
21sseq1d 3000 . . 3 (𝐴 = 𝐵 → ( 𝐴𝐴 𝐵𝐴))
3 sseq2 2995 . . 3 (𝐴 = 𝐵 → ( 𝐵𝐴 𝐵𝐵))
42, 3bitrd 181 . 2 (𝐴 = 𝐵 → ( 𝐴𝐴 𝐵𝐵))
5 df-tr 3883 . 2 (Tr 𝐴 𝐴𝐴)
6 df-tr 3883 . 2 (Tr 𝐵 𝐵𝐵)
74, 5, 63bitr4g 216 1 (𝐴 = 𝐵 → (Tr 𝐴 ↔ Tr 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102   = wceq 1259  wss 2945   cuni 3608  Tr wtr 3882
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-rex 2329  df-in 2952  df-ss 2959  df-uni 3609  df-tr 3883
This theorem is referenced by:  truni  3896  ordeq  4137  ordsucim  4254  ordom  4357
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