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

Theorem breqtri 3874
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 3859 . 2 (𝐴𝑅𝐵𝐴𝑅𝐶)
41, 3mpbi 144 1 𝐴𝑅𝐶
Colors of variables: wff set class
Syntax hints:   = wceq 1290   class class class wbr 3851
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622  df-un 3004  df-sn 3456  df-pr 3457  df-op 3459  df-br 3852
This theorem is referenced by:  breqtrri  3876  3brtr3i  3878  le9lt10  8957  9lt10  9061  sqrt2gt1lt2  10536  trireciplem  10948  cos1bnd  11104  cos2bnd  11105  cos01gt0  11107  sin4lt0  11111  z4even  11248  ex-fl  11918
  Copyright terms: Public domain W3C validator