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Theorem breqtri 3816
Description: Substitution of equal classes into a binary relation. (Contributed by NM, 5-Aug-1993.)
Hypotheses
Ref Expression
breqtr.1 𝐴𝑅𝐵
breqtr.2 𝐵 = 𝐶
Assertion
Ref Expression
breqtri 𝐴𝑅𝐶

Proof of Theorem breqtri
StepHypRef Expression
1 breqtr.1 . 2 𝐴𝑅𝐵
2 breqtr.2 . . 3 𝐵 = 𝐶
32breq2i 3801 . 2 (𝐴𝑅𝐵𝐴𝑅𝐶)
41, 3mpbi 143 1 𝐴𝑅𝐶
Colors of variables: wff set class
Syntax hints:   = wceq 1285   class class class wbr 3793
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 2064
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794
This theorem is referenced by:  breqtrri  3818  3brtr3i  3820  le9lt10  8584  9lt10  8688  sqrt2gt1lt2  10073  z4even  10460  ex-fl  10741
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