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Theorem breq1i 3771
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 3767 . 2 (𝐴 = 𝐵 → (𝐴𝑅𝐶𝐵𝑅𝐶))
31, 2ax-mp 7 1 (𝐴𝑅𝐶𝐵𝑅𝐶)
Colors of variables: wff set class
Syntax hints:  wb 98   = wceq 1243   class class class wbr 3764
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765
This theorem is referenced by:  eqbrtri  3783  brtpos0  5867  euen1  6282  euen1b  6283  2dom  6285  caucvgprprlemnbj  6791  caucvgprprlemmu  6793  caucvgprprlemaddq  6806  caucvgprprlem1  6807  gt0srpr  6833  caucvgsr  6886  pitonnlem1  6921  pitoregt0  6925  axprecex  6954  axpre-mulgt0  6961  axcaucvglemres  6973  lt0neg1  7463  le0neg1  7465  reclt1  7862  addltmul  8161  eluz2b1  8539  nn01to3  8552  xlt0neg1  8751  xle0neg1  8753  iccshftr  8862  iccshftl  8864  iccdil  8866  icccntr  8868  bernneq  9369
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