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Theorem breq 3766
 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 2101 . 2 (𝑅 = 𝑆 → (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑆))
2 df-br 3765 . 2 (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
3 df-br 3765 . 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 1243   ∈ wcel 1393  ⟨cop 3378   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-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-cleq 2033  df-clel 2036  df-br 3765 This theorem is referenced by:  breqi  3770  breqd  3775  poeq1  4036  soeq1  4052  frforeq1  4080  weeq1  4093  fveq1  5177  foeqcnvco  5430  f1eqcocnv  5431  isoeq2  5442  isoeq3  5443  ofreq  5715  shftfvalg  9419  shftfval  9422
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