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Theorem eqbrtrdi 4026
Description: A chained equality inference for a binary relation. (Contributed by NM, 12-Oct-1999.)
Hypotheses
Ref Expression
eqbrtrdi.1 (𝜑𝐴 = 𝐵)
eqbrtrdi.2 𝐵𝑅𝐶
Assertion
Ref Expression
eqbrtrdi (𝜑𝐴𝑅𝐶)

Proof of Theorem eqbrtrdi
StepHypRef Expression
1 eqbrtrdi.2 . 2 𝐵𝑅𝐶
2 eqbrtrdi.1 . . 3 (𝜑𝐴 = 𝐵)
32breq1d 3997 . 2 (𝜑 → (𝐴𝑅𝐶𝐵𝑅𝐶))
41, 3mpbiri 167 1 (𝜑𝐴𝑅𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = 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:  eqbrtrrdi  4027  pm54.43  7160  nn0ledivnn  9717  xltnegi  9785  leexp1a  10524  facwordi  10667  faclbnd3  10670  resqrexlemlo  10970  efap0  11633  dvds1  11806  en1top  12836  dvef  13447  rpabscxpbnd  13618  zabsle1  13659  trirec0  14041
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