Theorem breq 4130

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 2302	. 2 |- ( R = S -> ( <. A , B >. e. R <-> <. A , B >. e. S ) )
2		df-br 4129	. 2 |- ( A R B <-> <. A , B >. e. R )
3		df-br 4129	. 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 1402   e. wcel 2209   <.cop 3711   class class class wbr 4128

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 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-cleq 2231  df-clel 2234  df-br 4129

This theorem is referenced by:  breqi  4134  breqd  4139  poeq1  4442  soeq1  4458  frforeq1  4486  weeq1  4499  fveq1  5692  foeqcnvco  5989  f1eqcocnv  5990  isoeq2  6001  isoeq3  6002  ofreq  6299  supeq3  7323  papeq1  7602  tapeq1  7611  shftfvalg  11564  shftfval  11567  pw1nct  16950