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Theorem eqbrtrid 4038
Description: B chained equality inference for a binary relation. (Contributed by NM, 11-Oct-1999.)
Hypotheses
Ref Expression
eqbrtrid.1 𝐴 = 𝐵
eqbrtrid.2 (𝜑𝐵𝑅𝐶)
Assertion
Ref Expression
eqbrtrid (𝜑𝐴𝑅𝐶)

Proof of Theorem eqbrtrid
StepHypRef Expression
1 eqbrtrid.2 . 2 (𝜑𝐵𝑅𝐶)
2 eqbrtrid.1 . 2 𝐴 = 𝐵
3 eqid 2177 . 2 𝐶 = 𝐶
41, 2, 33brtr4g 4037 1 (𝜑𝐴𝑅𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353   class class class wbr 4003
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 2739  df-un 3133  df-sn 3598  df-pr 3599  df-op 3601  df-br 4004
This theorem is referenced by:  xp1en  6822  caucvgprlemm  7666  intqfrac2  10318  m1modge3gt1  10370  bernneq2  10641  reccn2ap  11320  eirraplem  11783  nno  11910  oddprmge3  12134  sqnprm  12135  4sqlem6  12380  oddennn  12392  strle2g  12565  strle3g  12566  1strstrg  12574  2strstrg  12576  rngstrg  12592  srngstrd  12603  lmodstrd  12621  ipsstrd  12633  topgrpstrd  12650  psmetge0  13801  reeff1olem  14162  cosq14gt0  14223  cosq34lt1  14241  ioocosf1o  14245  lgseisenlem1  14420  pwf1oexmid  14719  trilpolemeq1  14758
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