Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  breq2i GIF version

Theorem breq2i 3795
 Description: Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.)
Hypothesis
Ref Expression
breq1i.1 𝐴 = 𝐵
Assertion
Ref Expression
breq2i (𝐶𝑅𝐴𝐶𝑅𝐵)

Proof of Theorem breq2i
StepHypRef Expression
1 breq1i.1 . 2 𝐴 = 𝐵
2 breq2 3791 . 2 (𝐴 = 𝐵 → (𝐶𝑅𝐴𝐶𝑅𝐵))
31, 2ax-mp 7 1 (𝐶𝑅𝐴𝐶𝑅𝐵)
 Colors of variables: wff set class Syntax hints:   ↔ wb 103   = wceq 1285   class class class wbr 3787 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064 This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-sn 3406  df-pr 3407  df-op 3409  df-br 3788 This theorem is referenced by:  breqtri  3810  en1  6338  snnen2og  6384  1nen2  6386  pm54.43  6508  caucvgprprlemval  6929  caucvgprprlemmu  6936  caucvgsr  7029  pitonnlem1  7064  lt0neg2  7629  le0neg2  7631  negap0  7785  recexaplem2  7798  recgt1  8031  crap0  8091  addltmul  8323  nn0lt10b  8498  nn0lt2  8499  3halfnz  8514  xlt0neg2  8971  xle0neg2  8973  iccshftr  9081  iccshftl  9083  iccdil  9085  icccntr  9087  sizeen1  9812  cjap0  9921  abs00ap  10075  3dvdsdec  10398  3dvds2dec  10399  ndvdsi  10466  3prm  10643  prmfac1  10664
 Copyright terms: Public domain W3C validator