Theorem breq 3757

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

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