Theorem breq 4036

Description: Equality theorem for binary relations. (Contributed by NM, 4-Jun-1995.)

Assertion

Ref	Expression
breq	⊢ (𝑅 = 𝑆 → (𝐴𝑅𝐵 ↔ 𝐴𝑆𝐵))

Proof of Theorem breq

Step	Hyp	Ref	Expression
1		eleq2 2260	. 2 ⊢ (𝑅 = 𝑆 → (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑆))
2		df-br 4035	. 2 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
3		df-br 4035	. 2 ⊢ (𝐴𝑆𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑆)
4	1, 2, 3	3bitr4g 223	1 ⊢ (𝑅 = 𝑆 → (𝐴𝑅𝐵 ↔ 𝐴𝑆𝐵))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1364   ∈ wcel 2167  ⟨cop 3626   class class class wbr 4034

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-5 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-17 1540  ax-ial 1548  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-cleq 2189  df-clel 2192  df-br 4035

This theorem is referenced by:  breqi  4040  breqd  4045  poeq1  4335  soeq1  4351  frforeq1  4379  weeq1  4392  fveq1  5560  foeqcnvco  5840  f1eqcocnv  5841  isoeq2  5852  isoeq3  5853  ofreq  6143  supeq3  7065  tapeq1  7335  shftfvalg  11000  shftfval  11003  pw1nct  15734