Theorem breq 3795

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 2143	. 2 |- ( R = S -> ( <. A , B >. e. R <-> <. A , B >. e. S ) )
2		df-br 3794	. 2 |- ( A R B <-> <. A , B >. e. R )
3		df-br 3794	. 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 1285   e. wcel 1434   <.cop 3409   class class class wbr 3793

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 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-17 1460  ax-ial 1468  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-cleq 2075  df-clel 2078  df-br 3794

This theorem is referenced by:  breqi  3799  breqd  3804  poeq1  4062  soeq1  4078  frforeq1  4106  weeq1  4119  fveq1  5208  foeqcnvco  5461  f1eqcocnv  5462  isoeq2  5473  isoeq3  5474  ofreq  5746  supeq3  6462  shftfvalg  9844  shftfval  9847