Theorem breqi 4136

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 4132	. 2 ⊢ (𝑅 = 𝑆 → (𝐴𝑅𝐵 ↔ 𝐴𝑆𝐵))
3	1, 2	ax-mp 5	1 ⊢ (𝐴𝑅𝐵 ↔ 𝐴𝑆𝐵)

This proof depends on syntax axioms:   ↔ wb 105   = wceq 1402   class class class wbr 4130

This proof depends on 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 proof depends on definitions:  df-bi 117  df-cleq 2231  df-clel 2234  df-br 4131

This theorem is used by:  f1ompt  5859  brtpos2  6522  tfrexlem  6605  brdifun  6834  ltpiord  7686  ltxrlt  8391  ltxr  10177  xmeterval  15536