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Theorem treq 4002
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 3715 . . . 4 (𝐴 = 𝐵 𝐴 = 𝐵)
21sseq1d 3096 . . 3 (𝐴 = 𝐵 → ( 𝐴𝐴 𝐵𝐴))
3 sseq2 3091 . . 3 (𝐴 = 𝐵 → ( 𝐵𝐴 𝐵𝐵))
42, 3bitrd 187 . 2 (𝐴 = 𝐵 → ( 𝐴𝐴 𝐵𝐵))
5 df-tr 3997 . 2 (Tr 𝐴 𝐴𝐴)
6 df-tr 3997 . 2 (Tr 𝐵 𝐵𝐵)
74, 5, 63bitr4g 222 1 (𝐴 = 𝐵 → (Tr 𝐴 ↔ Tr 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1316  wss 3041   cuni 3706  Tr wtr 3996
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rex 2399  df-in 3047  df-ss 3054  df-uni 3707  df-tr 3997
This theorem is referenced by:  truni  4010  ordeq  4264  ordsucim  4386  ordom  4490  exmidonfinlem  7017
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