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Theorem breqtri 4012
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 3995 . 2 (𝐴𝑅𝐵𝐴𝑅𝐶)
41, 3mpbi 144 1 𝐴𝑅𝐶
Colors of variables: wff set class
Syntax hints:   = wceq 1348   class class class wbr 3987
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-sn 3587  df-pr 3588  df-op 3590  df-br 3988
This theorem is referenced by:  breqtrri  4014  3brtr3i  4016  le9lt10  9362  9lt10  9466  sqrt2gt1lt2  11006  trireciplem  11456  cos1bnd  11715  cos2bnd  11716  cos01gt0  11718  sin4lt0  11722  z4even  11868  coseq00topi  13515  sincos4thpi  13520  lgsdir2lem2  13689  lgsdir2lem3  13690  ex-fl  13725
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