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Theorem 3brtr3i 4032
Description: Substitution of equality into both sides of a binary relation. (Contributed by NM, 11-Aug-1999.)
Hypotheses
Ref Expression
3brtr3.1 𝐴𝑅𝐵
3brtr3.2 𝐴 = 𝐶
3brtr3.3 𝐵 = 𝐷
Assertion
Ref Expression
3brtr3i 𝐶𝑅𝐷

Proof of Theorem 3brtr3i
StepHypRef Expression
1 3brtr3.2 . . 3 𝐴 = 𝐶
2 3brtr3.1 . . 3 𝐴𝑅𝐵
31, 2eqbrtrri 4026 . 2 𝐶𝑅𝐵
4 3brtr3.3 . 2 𝐵 = 𝐷
53, 4breqtri 4028 1 𝐶𝑅𝐷
Colors of variables: wff set class
Syntax hints:   = wceq 1353   class class class wbr 4003
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-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-un 3133  df-sn 3598  df-pr 3599  df-op 3601  df-br 4004
This theorem is referenced by:  suplocsrlempr  7805  iap0  9140  ef01bndlem  11759
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