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Theorem eqbrtrid 4039
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 4038 1 (𝜑𝐴𝑅𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = 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:  xp1en  6823  caucvgprlemm  7667  intqfrac2  10319  m1modge3gt1  10371  bernneq2  10642  reccn2ap  11321  eirraplem  11784  nno  11911  oddprmge3  12135  sqnprm  12136  4sqlem6  12381  oddennn  12393  strle2g  12566  strle3g  12567  1strstrg  12575  2strstrg  12577  rngstrg  12593  srngstrd  12604  lmodstrd  12622  ipsstrd  12634  topgrpstrd  12651  psmetge0  13834  reeff1olem  14195  cosq14gt0  14256  cosq34lt1  14274  ioocosf1o  14278  lgseisenlem1  14453  pwf1oexmid  14752  trilpolemeq1  14791
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