Theorem xorass 1355

Description: \/_ is associative. (Contributed by FL, 22-Nov-2010.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Assertion

Ref	Expression
xorass	|- ( ( ( ph \/_ ps ) \/_ ch ) <-> ( ph \/_ ( ps \/_ ch ) ) )

Proof of Theorem xorass

Step	Hyp	Ref	Expression
1		biass 359	. . . . . 6 |- ( ( ( ph <-> ps ) <-> ch ) <-> ( ph <-> ( ps <-> ch ) ) )
2	1	notbii 296	. . . . 5 |- ( -. ( ( ph <-> ps ) <-> ch ) <-> -. ( ph <-> ( ps <-> ch ) ) )
3		nbbn 358	. . . . 5 |- ( ( -. ( ph <-> ps ) <-> ch ) <-> -. ( ( ph <-> ps ) <-> ch ) )
4		pm5.18 356	. . . . . 6 |- ( ( ph <-> ( ps <-> ch ) ) <-> -. ( ph <-> -. ( ps <-> ch ) ) )
5	4	con2bii 332	. . . . 5 |- ( ( ph <-> -. ( ps <-> ch ) ) <-> -. ( ph <-> ( ps <-> ch ) ) )
6	2, 3, 5	3bitr4i 277	. . . 4 |- ( ( -. ( ph <-> ps ) <-> ch ) <-> ( ph <-> -. ( ps <-> ch ) ) )
7		df-xor 1352	. . . . 5 |- ( ( ph \/_ ps ) <-> -. ( ph <-> ps ) )
8	7	bibi1i 314	. . . 4 |- ( ( ( ph \/_ ps ) <-> ch ) <-> ( -. ( ph <-> ps ) <-> ch ) )
9		df-xor 1352	. . . . 5 |- ( ( ps \/_ ch ) <-> -. ( ps <-> ch ) )
10	9	bibi2i 313	. . . 4 |- ( ( ph <-> ( ps \/_ ch ) ) <-> ( ph <-> -. ( ps <-> ch ) ) )
11	6, 8, 10	3bitr4i 277	. . 3 |- ( ( ( ph \/_ ps ) <-> ch ) <-> ( ph <-> ( ps \/_ ch ) ) )
12	11	notbii 296	. 2 |- ( -. ( ( ph \/_ ps ) <-> ch ) <-> -. ( ph <-> ( ps \/_ ch ) ) )
13		df-xor 1352	. 2 |- ( ( ( ph \/_ ps ) \/_ ch ) <-> -. ( ( ph \/_ ps ) <-> ch ) )
14		df-xor 1352	. 2 |- ( ( ph \/_ ( ps \/_ ch ) ) <-> -. ( ph <-> ( ps \/_ ch ) ) )
15	12, 13, 14	3bitr4i 277	1 |- ( ( ( ph \/_ ps ) \/_ ch ) <-> ( ph \/_ ( ps \/_ ch ) ) )

Syntax hints:   -. wn 3   <-> wb 184   \/_ wxo 1351

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 185  df-xor 1352

This theorem is referenced by:  hadass  1428