Theorem breq 4090

Description: Equality theorem for binary relations. (Contributed by NM, 4-Jun-1995.)

Assertion

Ref	Expression
breq	|- ( R = S -> ( A R B <-> A S B ) )

Proof of Theorem breq

Step	Hyp	Ref	Expression
1		eleq2 2295	. 2 |- ( R = S -> ( <. A , B >. e. R <-> <. A , B >. e. S ) )
2		df-br 4089	. 2 |- ( A R B <-> <. A , B >. e. R )
3		df-br 4089	. 2 |- ( A S B <-> <. A , B >. e. S )
4	1, 2, 3	3bitr4g 223	1 |- ( R = S -> ( A R B <-> A S B ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1397   e. wcel 2202   <.cop 3672   class class class wbr 4088

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 1495  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-4 1558  ax-17 1574  ax-ial 1582  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-cleq 2224  df-clel 2227  df-br 4089

This theorem is referenced by:  breqi  4094  breqd  4099  poeq1  4396  soeq1  4412  frforeq1  4440  weeq1  4453  fveq1  5639  foeqcnvco  5934  f1eqcocnv  5935  isoeq2  5946  isoeq3  5947  ofreq  6242  supeq3  7192  tapeq1  7474  shftfvalg  11399  shftfval  11402  pw1nct  16663