Theorem breq 3939

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 2204	. 2 |- ( R = S -> ( <. A , B >. e. R <-> <. A , B >. e. S ) )
2		df-br 3938	. 2 |- ( A R B <-> <. A , B >. e. R )
3		df-br 3938	. 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 1332   e. wcel 1481   <.cop 3535   class class class wbr 3937

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 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-17 1507  ax-ial 1515  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-cleq 2133  df-clel 2136  df-br 3938

This theorem is referenced by:  breqi  3943  breqd  3948  poeq1  4229  soeq1  4245  frforeq1  4273  weeq1  4286  fveq1  5428  foeqcnvco  5699  f1eqcocnv  5700  isoeq2  5711  isoeq3  5712  ofreq  5993  supeq3  6885  shftfvalg  10622  shftfval  10625  pw1nct  13371