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Theorem breq1i 4011
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 4007 . 2 (𝐴 = 𝐵 → (𝐴𝑅𝐶𝐵𝑅𝐶))
31, 2ax-mp 5 1 (𝐴𝑅𝐶𝐵𝑅𝐶)
Colors of variables: wff set class
Syntax hints:  wb 105   = wceq 1353   class class class wbr 4004
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2740  df-un 3134  df-sn 3599  df-pr 3600  df-op 3602  df-br 4005
This theorem is referenced by:  eqbrtri  4025  brtpos0  6253  euen1  6802  euen1b  6803  2dom  6805  infglbti  7024  pr2nelem  7190  caucvgprprlemnbj  7692  caucvgprprlemmu  7694  caucvgprprlemaddq  7707  caucvgprprlem1  7708  gt0srpr  7747  caucvgsr  7801  mappsrprg  7803  map2psrprg  7804  pitonnlem1  7844  pitoregt0  7848  axprecex  7879  axpre-mulgt0  7886  axcaucvglemres  7898  lt0neg1  8425  le0neg1  8427  reclt1  8853  addltmul  9155  eluz2b1  9601  nn01to3  9617  xlt0neg1  9838  xle0neg1  9840  iccshftr  9994  iccshftl  9996  iccdil  9998  icccntr  10000  bernneq  10641  cbvsum  11368  expcnv  11512  cbvprod  11566  oddge22np1  11886  nn0o1gt2  11910  isprm3  12118  dvdsnprmd  12125  pw2dvdslemn  12165  txmetcnp  14021  sincosq1sgn  14250  sincosq3sgn  14252  sincosq4sgn  14253  logrpap0b  14300
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