Theorem brsymdif 4477

Description: The binary relationship of a symmetric difference. (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 4421	. 2 |- ( A ( R /_\ S ) B <-> <. A , B >. e. ( R /_\ S ) )
2		elsymdif 3698	. . 3 |- ( <. A , B >. e. ( R /_\ S ) <-> -. ( <. A , B >. e. R <-> <. A , B >. e. S ) )
3		df-br 4421	. . . 4 |- ( A R B <-> <. A , B >. e. R )
4		df-br 4421	. . . 4 |- ( A S B <-> <. A , B >. e. S )
5	3, 4	bibi12i 316	. . 3 |- ( ( A R B <-> A S B ) <-> ( <. A , B >. e. R <-> <. A , B >. e. S ) )
6	2, 5	xchbinxr 312	. 2 |- ( <. A , B >. e. ( R /_\ S ) <-> -. ( A R B <-> A S B ) )
7	1, 6	bitri 252	1 |- ( A ( R /_\ S ) B <-> -. ( A R B <-> A S B ) )

Syntax hints:   -. wn 3   <-> wb 187   e. wcel 1868   /_\ csymdif 3692   <.cop 4002   class class class wbr 4420

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-v 3083  df-dif 3439  df-un 3441  df-symdif 3693  df-br 4421

This theorem is referenced by:  brtxpsd  30654