Theorem breqi 4646

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 4642	. 2 ⊢ (R = S → (ARB ↔ ASB))
3	1, 2	ax-mp 5	1 ⊢ (ARB ↔ ASB)

Syntax hints:   ↔ wb 176   = wceq 1642   class class class wbr 4640

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-cleq 2346  df-clel 2349  df-br 4641

This theorem is referenced by:  brres  4950  trtxp  5782  brtxp  5784  brimage  5794  qrpprod  5837  brpprod  5840  fnfullfunlem1  5857  ersymtr  5933  porta  5934  sopc  5935  weds  5939  brltc  6115  nchoicelem8  6297  nchoicelem19  6308