Theorem breq 3740

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 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)
4	1, 2, 3	3bitr4g 212	1 ⊢ (𝑅 = 𝑆 → (A𝑅B ↔ A𝑆B))

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