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Theorem syl5breqr 3828
 Description: B chained equality inference for a binary relation. (Contributed by NM, 24-Apr-2005.)
Hypotheses
Ref Expression
syl5breqr.1 𝐴𝑅𝐵
syl5breqr.2 (𝜑𝐶 = 𝐵)
Assertion
Ref Expression
syl5breqr (𝜑𝐴𝑅𝐶)

Proof of Theorem syl5breqr
StepHypRef Expression
1 syl5breqr.1 . 2 𝐴𝑅𝐵
2 syl5breqr.2 . . 3 (𝜑𝐶 = 𝐵)
32eqcomd 2061 . 2 (𝜑𝐵 = 𝐶)
41, 3syl5breq 3827 1 (𝜑𝐴𝑅𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1259   class class class wbr 3792 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2950  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793 This theorem is referenced by:  bernneq  9537
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