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Theorem breqtri 3948
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 3932 . 2 (𝐴𝑅𝐵𝐴𝑅𝐶)
41, 3mpbi 144 1 𝐴𝑅𝐶
Colors of variables: wff set class
Syntax hints:   = wceq 1331   class class class wbr 3924
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-un 3070  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925
This theorem is referenced by:  breqtrri  3950  3brtr3i  3952  le9lt10  9201  9lt10  9305  sqrt2gt1lt2  10814  trireciplem  11262  cos1bnd  11455  cos2bnd  11456  cos01gt0  11458  sin4lt0  11462  z4even  11602  coseq00topi  12905  sincos4thpi  12910  ex-fl  12926
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