Theorem brdif

Description: The difference of two binary relations. (Contributed by Scott Fenton, 11-Apr-2011)

	Ref	Expression
Assertion	brdif	|- ( A ( R \ S ) B <-> ( A R B /\ -. A S B ) )

Proof

Step	Hyp	Ref	Expression
1		eldif	|- ( <. A , B >. e. ( R \ S ) <-> ( <. A , B >. e. R /\ -. <. A , B >. e. S ) )
2		df-br	|- ( A ( R \ S ) B <-> <. A , B >. e. ( R \ S ) )
3		df-br	|- ( A R B <-> <. A , B >. e. R )
4		df-br	|- ( A S B <-> <. A , B >. e. S )
5	4	notbii	|- ( -. A S B <-> -. <. A , B >. e. S )
6	3 5	anbi12i	|- ( ( A R B /\ -. A S B ) <-> ( <. A , B >. e. R /\ -. <. A , B >. e. S ) )
7	1 2 6	3bitr4i	|- ( A ( R \ S ) B <-> ( A R B /\ -. A S B ) )