Theorem brsymdif

Description: Characterization of the symmetric difference of two binary relations. (Contributed by Scott Fenton, 11-Apr-2012)

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

Proof

Step	Hyp	Ref	Expression
1		df-br	|- ( A ( R /_\ S ) B <-> <. A , B >. e. ( R /_\ S ) )
2		elsymdif	|- ( <. A , B >. e. ( R /_\ S ) <-> -. ( <. A , B >. e. R <-> <. A , B >. e. S ) )
3		df-br	|- ( A R B <-> <. A , B >. e. R )
4		df-br	|- ( A S B <-> <. A , B >. e. S )
5	3 4	bibi12i	|- ( ( A R B <-> A S B ) <-> ( <. A , B >. e. R <-> <. A , B >. e. S ) )
6	2 5	xchbinxr	|- ( <. A , B >. e. ( R /_\ S ) <-> -. ( A R B <-> A S B ) )
7	1 6	bitri	|- ( A ( R /_\ S ) B <-> -. ( A R B <-> A S B ) )