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Theorem breq 3740
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 2083 . 2 (𝑅 = 𝑆 → (⟨A, B 𝑅 ↔ ⟨A, B 𝑆))
2 df-br 3739 . 2 (A𝑅B ↔ ⟨A, B 𝑅)
3 df-br 3739 . 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 1228   wcel 1374  cop 3353   class class class wbr 3738
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 1316  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-4 1381  ax-17 1400  ax-ial 1409  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-cleq 2015  df-clel 2018  df-br 3739
This theorem is referenced by:  breqi  3744  breqd  3749  poeq1  4010  soeq1  4026  fveq1  5102  foeqcnvco  5355  f1eqcocnv  5356  isoeq2  5367  isoeq3  5368  ofreq  5638
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