Theorem breq 3787

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 2142	. 2 |- ( R = S -> ( <. A , B >. e. R <-> <. A , B >. e. S ) )
2		df-br 3786	. 2 |- ( A R B <-> <. A , B >. e. R )
3		df-br 3786	. 2 |- ( A S B <-> <. A , B >. e. S )
4	1, 2, 3	3bitr4g 221	1 |- ( R = S -> ( A R B <-> A S B ) )

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1284   e. wcel 1433   <.cop 3401   class class class wbr 3785

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-17 1459  ax-ial 1467  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-cleq 2074  df-clel 2077  df-br 3786

This theorem is referenced by:  breqi  3791  breqd  3796  poeq1  4054  soeq1  4070  frforeq1  4098  weeq1  4111  fveq1  5197  foeqcnvco  5450  f1eqcocnv  5451  isoeq2  5462  isoeq3  5463  ofreq  5735  supeq3  6403  shftfvalg  9706  shftfval  9709