Theorem assxor 14279

Description: (+) is associative. (The proof was shortened by Andrew Salmon, 8-Jun-2011.)

Assertion

Ref	Expression
assxor	|- (((ph(+)ps)(+)ch) <-> (ph(+)(ps(+)ch)))

Proof of Theorem assxor

Step	Hyp	Ref	Expression
1		biass 816	. . . . . 6 |- (((ph <-> ps) <-> ch) <-> (ph <-> (ps <-> ch)))
2	1	notbii 204	. . . . 5 |- (-. ((ph <-> ps) <-> ch) <-> -. (ph <-> (ps <-> ch)))
3		nbbn 724	. . . . 5 |- ((-. (ph <-> ps) <-> ch) <-> -. ((ph <-> ps) <-> ch))
4		pm5.18 722	. . . . . 6 |- ((ph <-> (ps <-> ch)) <-> -. (ph <-> -. (ps <-> ch)))
5	4	con2bii 238	. . . . 5 |- ((ph <-> -. (ps <-> ch)) <-> -. (ph <-> (ps <-> ch)))
6	2, 3, 5	3bitr4i 200	. . . 4 |- ((-. (ph <-> ps) <-> ch) <-> (ph <-> -. (ps <-> ch)))
7		df-xor 14278	. . . . 5 |- ((ph(+)ps) <-> -. (ph <-> ps))
8	7	bibi1i 671	. . . 4 |- (((ph(+)ps) <-> ch) <-> (-. (ph <-> ps) <-> ch))
9		df-xor 14278	. . . . 5 |- ((ps(+)ch) <-> -. (ps <-> ch))
10	9	bibi2i 669	. . . 4 |- ((ph <-> (ps(+)ch)) <-> (ph <-> -. (ps <-> ch)))
11	6, 8, 10	3bitr4i 200	. . 3 |- (((ph(+)ps) <-> ch) <-> (ph <-> (ps(+)ch)))
12	11	notbii 204	. 2 |- (-. ((ph(+)ps) <-> ch) <-> -. (ph <-> (ps(+)ch)))
13		df-xor 14278	. 2 |- (((ph(+)ps)(+)ch) <-> -. ((ph(+)ps) <-> ch))
14		df-xor 14278	. 2 |- ((ph(+)(ps(+)ch)) <-> -. (ph <-> (ps(+)ch)))
15	12, 13, 14	3bitr4i 200	1 |- (((ph(+)ps)(+)ch) <-> (ph(+)(ps(+)ch)))

Syntax hints:  -. wn 2   <-> wb 163  (+)wxo 14277

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7

This theorem depends on definitions:  df-bi 164  df-an 242  df-xor 14278

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