Theorem xorass 1308

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 348	. . . . . 6 |- (((ph <-> ps) <-> ch) <-> (ph <-> (ps <-> ch)))
2	1	notbii 287	. . . . 5 |- (-. ((ph <-> ps) <-> ch) <-> -. (ph <-> (ps <-> ch)))
3		nbbn 347	. . . . 5 |- ((-. (ph <-> ps) <-> ch) <-> -. ((ph <-> ps) <-> ch))
4		pm5.18 345	. . . . . 6 |- ((ph <-> (ps <-> ch)) <-> -. (ph <-> -. (ps <-> ch)))
5	4	con2bii 322	. . . . 5 |- ((ph <-> -. (ps <-> ch)) <-> -. (ph <-> (ps <-> ch)))
6	2, 3, 5	3bitr4i 268	. . . 4 |- ((-. (ph <-> ps) <-> ch) <-> (ph <-> -. (ps <-> ch)))
7		df-xor 1305	. . . . 5 |- ((ph \/_ ps) <-> -. (ph <-> ps))
8	7	bibi1i 305	. . . 4 |- (((ph \/_ ps) <-> ch) <-> (-. (ph <-> ps) <-> ch))
9		df-xor 1305	. . . . 5 |- ((ps \/_ ch) <-> -. (ps <-> ch))
10	9	bibi2i 304	. . . 4 |- ((ph <-> (ps \/_ ch)) <-> (ph <-> -. (ps <-> ch)))
11	6, 8, 10	3bitr4i 268	. . 3 |- (((ph \/_ ps) <-> ch) <-> (ph <-> (ps \/_ ch)))
12	11	notbii 287	. 2 |- (-. ((ph \/_ ps) <-> ch) <-> -. (ph <-> (ps \/_ ch)))
13		df-xor 1305	. 2 |- (((ph \/_ ps) \/_ ch) <-> -. ((ph \/_ ps) <-> ch))
14		df-xor 1305	. 2 |- ((ph \/_ (ps \/_ ch)) <-> -. (ph <-> (ps \/_ ch)))
15	12, 13, 14	3bitr4i 268	1 |- (((ph \/_ ps) \/_ ch) <-> (ph \/_ (ps \/_ ch)))

Syntax hints:  -. wn 3   <-> wb 176   \/_ wxo 1304

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 177  df-xor 1305

This theorem is referenced by:  hadass  1386