Theorem breqi 4050

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 4046	. 2 ⊢ (𝑅 = 𝑆 → (𝐴𝑅𝐵 ↔ 𝐴𝑆𝐵))
3	1, 2	ax-mp 5	1 ⊢ (𝐴𝑅𝐵 ↔ 𝐴𝑆𝐵)

Syntax hints:   ↔ wb 105   = wceq 1373   class class class wbr 4044

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 1470  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-4 1533  ax-17 1549  ax-ial 1557  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-cleq 2198  df-clel 2201  df-br 4045

This theorem is referenced by:  f1ompt  5731  brtpos2  6337  tfrexlem  6420  brdifun  6647  ltpiord  7432  ltxrlt  8138  ltxr  9897  xmeterval  14907