Theorem breqi 3770

Description: Equality inference for binary relations. (Contributed by NM, 19-Feb-2005.)

Hypothesis

Ref	Expression
breqi.1	⊢ 𝑅 = 𝑆

Assertion

Ref	Expression
breqi	⊢ (𝐴𝑅𝐵 ↔ 𝐴𝑆𝐵)

Proof of Theorem breqi

Step	Hyp	Ref	Expression
1		breqi.1	. 2 ⊢ 𝑅 = 𝑆
2		breq 3766	. 2 ⊢ (𝑅 = 𝑆 → (𝐴𝑅𝐵 ↔ 𝐴𝑆𝐵))
3	1, 2	ax-mp 7	1 ⊢ (𝐴𝑅𝐵 ↔ 𝐴𝑆𝐵)

Syntax hints:   ↔ wb 98   = wceq 1243   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:  f1ompt  5320  brtpos2  5866  tfrexlem  5948  brdifun  6133  ltpiord  6417  ltxrlt  7085  ltxr  8695