Theorem brsymdif 4711

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

Assertion

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

Proof of Theorem brsymdif

Step	Hyp	Ref	Expression
1		df-br 4654	. 2 |- ( A ( R /_\ S ) B <-> <. A , B >. e. ( R /_\ S ) )
2		elsymdif 3849	. . 3 |- ( <. A , B >. e. ( R /_\ S ) <-> -. ( <. A , B >. e. R <-> <. A , B >. e. S ) )
3		df-br 4654	. . . 4 |- ( A R B <-> <. A , B >. e. R )
4		df-br 4654	. . . 4 |- ( A S B <-> <. A , B >. e. S )
5	3, 4	bibi12i 329	. . 3 |- ( ( A R B <-> A S B ) <-> ( <. A , B >. e. R <-> <. A , B >. e. S ) )
6	2, 5	xchbinxr 325	. 2 |- ( <. A , B >. e. ( R /_\ S ) <-> -. ( A R B <-> A S B ) )
7	1, 6	bitri 264	1 |- ( A ( R /_\ S ) B <-> -. ( A R B <-> A S B ) )

Syntax hints:   -. wn 3   <-> wb 196   e. wcel 1990   /_\ csymdif 3843   <.cop 4183   class class class wbr 4653

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-dif 3577  df-un 3579  df-symdif 3844  df-br 4654

This theorem is referenced by:  brtxpsd  32001