Theorem brsymdif 4462

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 4406	. 2 |- ( A ( R /_\ S ) B <-> <. A , B >. e. ( R /_\ S ) )
2		elsymdif 3670	. . 3 |- ( <. A , B >. e. ( R /_\ S ) <-> -. ( <. A , B >. e. R <-> <. A , B >. e. S ) )
3		df-br 4406	. . . 4 |- ( A R B <-> <. A , B >. e. R )
4		df-br 4406	. . . 4 |- ( A S B <-> <. A , B >. e. S )
5	3, 4	bibi12i 317	. . 3 |- ( ( A R B <-> A S B ) <-> ( <. A , B >. e. R <-> <. A , B >. e. S ) )
6	2, 5	xchbinxr 313	. 2 |- ( <. A , B >. e. ( R /_\ S ) <-> -. ( A R B <-> A S B ) )
7	1, 6	bitri 253	1 |- ( A ( R /_\ S ) B <-> -. ( A R B <-> A S B ) )

Syntax hints:   -. wn 3   <-> wb 188   e. wcel 1889   /_\ csymdif 3664   <.cop 3976   class class class wbr 4405

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-v 3049  df-dif 3409  df-un 3411  df-symdif 3665  df-br 4406

This theorem is referenced by:  brtxpsd  30673