Theorem breq 4133

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 4132	. 2 |- ( A R B <-> <. A , B >. e. R )
3		df-br 4132	. 2 |- ( A S B <-> <. A , B >. e. S )
4	1, 2, 3	3bitr4g 223	1 |- ( R = S -> ( A R B <-> A S B ) )

This proof depends on syntax axioms:   -> wi 4   <-> wb 105   = wceq 1402   e. wcel 2209   <.cop 3712   class class class wbr 4131

This proof depends on 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 proof depends on definitions:  df-bi 117  df-cleq 2231  df-clel 2234  df-br 4132

This theorem is used by:  breqi  4137  breqd  4142  poeq1  4446  soeq1  4462  frforeq1  4490  weeq1  4503  fveq1  5697  foeqcnvco  5999  f1eqcocnv  6000  isoeq2  6011  isoeq3  6012  ofreq  6309  supeq3  7336  papeq1  7615  tapeq1  7624  shftfvalg  11605  shftfval  11608  pw1nct  17238