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Theorem syl5eqbrr 3798
Description: B chained equality inference for a binary relation. (Contributed by NM, 17-Sep-2004.)
Hypotheses
Ref Expression
syl5eqbrr.1 𝐵 = 𝐴
syl5eqbrr.2 (𝜑𝐵𝑅𝐶)
Assertion
Ref Expression
syl5eqbrr (𝜑𝐴𝑅𝐶)

Proof of Theorem syl5eqbrr
StepHypRef Expression
1 syl5eqbrr.2 . 2 (𝜑𝐵𝑅𝐶)
2 syl5eqbrr.1 . 2 𝐵 = 𝐴
3 eqid 2040 . 2 𝐶 = 𝐶
41, 2, 33brtr3g 3795 1 (𝜑𝐴𝑅𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = 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:  enpr1g  6278  recexprlem1ssl  6731  addgt0  7443  addgegt0  7444  addgtge0  7445  addge0  7446  expge1  9292
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