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Theorem breqtrrid 4038
Description: B chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.)
Hypotheses
Ref Expression
breqtrrid.1 𝐴𝑅𝐵
breqtrrid.2 (𝜑𝐶 = 𝐵)
Assertion
Ref Expression
breqtrrid (𝜑𝐴𝑅𝐶)

Proof of Theorem breqtrrid
StepHypRef Expression
1 breqtrrid.1 . 2 𝐴𝑅𝐵
2 breqtrrid.2 . . 3 (𝜑𝐶 = 𝐵)
32eqcomd 2183 . 2 (𝜑𝐵 = 𝐶)
41, 3breqtrid 4037 1 (𝜑𝐴𝑅𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353   class class class wbr 4000
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 3597  df-pr 3598  df-op 3600  df-br 4001
This theorem is referenced by:  xsubge0  9865  xposdif  9866  bernneq  10623  pcge0  12292  rpabscxpbnd  14019  lgsdir2lem2  14090  trilpolemclim  14433  trilpolemlt1  14438  nconstwlpolemgt0  14460
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