Theorem breq 3847

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 2151	. 2 |- ( R = S -> ( <. A , B >. e. R <-> <. A , B >. e. S ) )
2		df-br 3846	. 2 |- ( A R B <-> <. A , B >. e. R )
3		df-br 3846	. 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 1289   e. wcel 1438   <.cop 3449   class class class wbr 3845

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 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-17 1464  ax-ial 1472  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-cleq 2081  df-clel 2084  df-br 3846

This theorem is referenced by:  breqi  3851  breqd  3856  poeq1  4126  soeq1  4142  frforeq1  4170  weeq1  4183  fveq1  5304  foeqcnvco  5569  f1eqcocnv  5570  isoeq2  5581  isoeq3  5582  ofreq  5859  supeq3  6683  shftfvalg  10248  shftfval  10251