Theorem breqi 4089

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 4085	. 2 ⊢ (𝑅 = 𝑆 → (𝐴𝑅𝐵 ↔ 𝐴𝑆𝐵))
3	1, 2	ax-mp 5	1 ⊢ (𝐴𝑅𝐵 ↔ 𝐴𝑆𝐵)

Syntax hints:   ↔ wb 105   = wceq 1395   class class class wbr 4083

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 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-ial 1580  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-cleq 2222  df-clel 2225  df-br 4084

This theorem is referenced by:  f1ompt  5786  brtpos2  6397  tfrexlem  6480  brdifun  6707  ltpiord  7506  ltxrlt  8212  ltxr  9971  xmeterval  15109