Theorem xorass 1408

Description: \/_ is associative. (Contributed by FL, 22-Nov-2010.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Proof shortened by Wolf Lammen, 20-Jun-2020.)

Assertion

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

Proof of Theorem xorass

Step	Hyp	Ref	Expression
1		xor3 359	. . 3 |- ( -. ( ph <-> ( ps \/_ ch ) ) <-> ( ph <-> -. ( ps \/_ ch ) ) )
2		biass 361	. . . 4 |- ( ( ( ph <-> ps ) <-> ch ) <-> ( ph <-> ( ps <-> ch ) ) )
3		xnor 1406	. . . . 5 |- ( ( ph <-> ps ) <-> -. ( ph \/_ ps ) )
4	3	bibi1i 316	. . . 4 |- ( ( ( ph <-> ps ) <-> ch ) <-> ( -. ( ph \/_ ps ) <-> ch ) )
5		xnor 1406	. . . . 5 |- ( ( ps <-> ch ) <-> -. ( ps \/_ ch ) )
6	5	bibi2i 315	. . . 4 |- ( ( ph <-> ( ps <-> ch ) ) <-> ( ph <-> -. ( ps \/_ ch ) ) )
7	2, 4, 6	3bitr3i 279	. . 3 |- ( ( -. ( ph \/_ ps ) <-> ch ) <-> ( ph <-> -. ( ps \/_ ch ) ) )
8		nbbn 360	. . 3 |- ( ( -. ( ph \/_ ps ) <-> ch ) <-> -. ( ( ph \/_ ps ) <-> ch ) )
9	1, 7, 8	3bitr2ri 278	. 2 |- ( -. ( ( ph \/_ ps ) <-> ch ) <-> -. ( ph <-> ( ps \/_ ch ) ) )
10		df-xor 1405	. 2 |- ( ( ( ph \/_ ps ) \/_ ch ) <-> -. ( ( ph \/_ ps ) <-> ch ) )
11		df-xor 1405	. 2 |- ( ( ph \/_ ( ps \/_ ch ) ) <-> -. ( ph <-> ( ps \/_ ch ) ) )
12	9, 10, 11	3bitr4i 281	1 |- ( ( ( ph \/_ ps ) \/_ ch ) <-> ( ph \/_ ( ps \/_ ch ) ) )

Syntax hints:   -. wn 3   <-> wb 188   \/_ wxo 1404

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 189  df-xor 1405

This theorem is referenced by:  hadass  1499