ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  eqbrtrdi GIF version

Theorem eqbrtrdi 4028
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 3999 . 2 (𝜑 → (𝐴𝑅𝐶𝐵𝑅𝐶))
41, 3mpbiri 167 1 (𝜑𝐴𝑅𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1348   class class class wbr 3989
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 3589  df-pr 3590  df-op 3592  df-br 3990
This theorem is referenced by:  eqbrtrrdi  4029  pm54.43  7167  nn0ledivnn  9724  xltnegi  9792  leexp1a  10531  facwordi  10674  faclbnd3  10677  resqrexlemlo  10977  efap0  11640  dvds1  11813  en1top  12871  dvef  13482  rpabscxpbnd  13653  zabsle1  13694  trirec0  14076
  Copyright terms: Public domain W3C validator