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Theorem breq 3757
Description: Equality theorem for binary relations. (Contributed by NM, 4-Jun-1995.)
Assertion
Ref Expression
breq (𝑅 = 𝑆 → (A𝑅BA𝑆B))

Proof of Theorem breq
StepHypRef Expression
1 eleq2 2098 . 2 (𝑅 = 𝑆 → (⟨A, B 𝑅 ↔ ⟨A, B 𝑆))
2 df-br 3756 . 2 (A𝑅B ↔ ⟨A, B 𝑅)
3 df-br 3756 . 2 (A𝑆B ↔ ⟨A, B 𝑆)
41, 2, 33bitr4g 212 1 (𝑅 = 𝑆 → (A𝑅BA𝑆B))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1242   wcel 1390  cop 3370   class class class wbr 3755
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-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-17 1416  ax-ial 1424  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-cleq 2030  df-clel 2033  df-br 3756
This theorem is referenced by:  breqi  3761  breqd  3766  poeq1  4027  soeq1  4043  fveq1  5120  foeqcnvco  5373  f1eqcocnv  5374  isoeq2  5385  isoeq3  5386  ofreq  5657
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