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Theorem eqbrtri 3824
 Description: Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
Hypotheses
Ref Expression
eqbrtr.1 𝐴 = 𝐵
eqbrtr.2 𝐵𝑅𝐶
Assertion
Ref Expression
eqbrtri 𝐴𝑅𝐶

Proof of Theorem eqbrtri
StepHypRef Expression
1 eqbrtr.2 . 2 𝐵𝑅𝐶
2 eqbrtr.1 . . 3 𝐴 = 𝐵
32breq1i 3812 . 2 (𝐴𝑅𝐶𝐵𝑅𝐶)
41, 3mpbir 144 1 𝐴𝑅𝐶
 Colors of variables: wff set class Syntax hints:   = wceq 1285   class class class wbr 3805 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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065 This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-un 2986  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806 This theorem is referenced by:  eqbrtrri  3826  3brtr4i  3833  neg1lt0  8266  halflt1  8367  3halfnz  8577  declei  8645  numlti  8646  faclbnd3  9819  1nprm  10703  znnen  10818  ex-fl  10823
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