Theorem breq 4127

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 2302	. 2 ⊢ (𝑅 = 𝑆 → (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑆))
2		df-br 4126	. 2 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
3		df-br 4126	. 2 ⊢ (𝐴𝑆𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑆)
4	1, 2, 3	3bitr4g 223	1 ⊢ (𝑅 = 𝑆 → (𝐴𝑅𝐵 ↔ 𝐴𝑆𝐵))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1402   ∈ wcel 2209  ⟨cop 3708   class class class wbr 4125

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 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-cleq 2231  df-clel 2234  df-br 4126

This theorem is referenced by:  breqi  4131  breqd  4136  poeq1  4439  soeq1  4455  frforeq1  4483  weeq1  4496  fveq1  5689  foeqcnvco  5986  f1eqcocnv  5987  isoeq2  5998  isoeq3  5999  ofreq  6296  supeq3  7320  papeq1  7599  tapeq1  7608  shftfvalg  11561  shftfval  11564  pw1nct  16947