Theorem breq 4036

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 2260	. 2 |- ( R = S -> ( <. A , B >. e. R <-> <. A , B >. e. S ) )
2		df-br 4035	. 2 |- ( A R B <-> <. A , B >. e. R )
3		df-br 4035	. 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 1364   e. wcel 2167   <.cop 3626   class class class wbr 4034

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 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-17 1540  ax-ial 1548  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-cleq 2189  df-clel 2192  df-br 4035

This theorem is referenced by:  breqi  4040  breqd  4045  poeq1  4335  soeq1  4351  frforeq1  4379  weeq1  4392  fveq1  5558  foeqcnvco  5838  f1eqcocnv  5839  isoeq2  5850  isoeq3  5851  ofreq  6141  supeq3  7058  tapeq1  7322  shftfvalg  10986  shftfval  10989  pw1nct  15676