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Theorem breq1i 3812
Description: Equality inference for a binary relation. (Contributed by NM, 8-Feb-1996.)
Hypothesis
Ref Expression
breq1i.1 𝐴 = 𝐵
Assertion
Ref Expression
breq1i (𝐴𝑅𝐶𝐵𝑅𝐶)

Proof of Theorem breq1i
StepHypRef Expression
1 breq1i.1 . 2 𝐴 = 𝐵
2 breq1 3808 . 2 (𝐴 = 𝐵 → (𝐴𝑅𝐶𝐵𝑅𝐶))
31, 2ax-mp 7 1 (𝐴𝑅𝐶𝐵𝑅𝐶)
Colors of variables: wff set class
Syntax hints:  wb 103   = wceq 1285   class class class wbr 3805
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 2065
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-un 2986  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806
This theorem is referenced by:  eqbrtri  3824  brtpos0  5921  euen1  6370  euen1b  6371  2dom  6373  infglbti  6532  pr2nelem  6571  caucvgprprlemnbj  6997  caucvgprprlemmu  6999  caucvgprprlemaddq  7012  caucvgprprlem1  7013  gt0srpr  7039  caucvgsr  7092  pitonnlem1  7127  pitoregt0  7131  axprecex  7160  axpre-mulgt0  7167  axcaucvglemres  7179  lt0neg1  7691  le0neg1  7693  reclt1  8093  addltmul  8386  eluz2b1  8821  nn01to3  8835  xlt0neg1  9033  xle0neg1  9035  iccshftr  9144  iccshftl  9146  iccdil  9148  icccntr  9150  bernneq  9742  oddge22np1  10488  nn0o1gt2  10512  isprm3  10707  dvdsnprmd  10714  pw2dvdslemn  10750
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