Theorem breq 3931

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 2203	. 2 |- ( R = S -> ( <. A , B >. e. R <-> <. A , B >. e. S ) )
2		df-br 3930	. 2 |- ( A R B <-> <. A , B >. e. R )
3		df-br 3930	. 2 |- ( A S B <-> <. A , B >. e. S )
4	1, 2, 3	3bitr4g 222	1 |- ( R = S -> ( A R B <-> A S B ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1331   e. wcel 1480   <.cop 3530   class class class wbr 3929

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-cleq 2132  df-clel 2135  df-br 3930

This theorem is referenced by:  breqi  3935  breqd  3940  poeq1  4221  soeq1  4237  frforeq1  4265  weeq1  4278  fveq1  5420  foeqcnvco  5691  f1eqcocnv  5692  isoeq2  5703  isoeq3  5704  ofreq  5985  supeq3  6877  shftfvalg  10590  shftfval  10593