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Metamath Proof Explorer

Theorem xorass

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

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

Proof

Step Hyp Ref Expression
1 xor3 
 |-  ( -. ( ph <-> ( ps \/_ ch ) ) <-> ( ph <-> -. ( ps \/_ ch ) ) )
2 biass 
 |-  ( ( ( ph <-> ps ) <-> ch ) <-> ( ph <-> ( ps <-> ch ) ) )
3 xnor 
 |-  ( ( ph <-> ps ) <-> -. ( ph \/_ ps ) )
4 3 bibi1i 
 |-  ( ( ( ph <-> ps ) <-> ch ) <-> ( -. ( ph \/_ ps ) <-> ch ) )
5 xnor 
 |-  ( ( ps <-> ch ) <-> -. ( ps \/_ ch ) )
6 5 bibi2i 
 |-  ( ( ph <-> ( ps <-> ch ) ) <-> ( ph <-> -. ( ps \/_ ch ) ) )
7 2 4 6 3bitr3i 
 |-  ( ( -. ( ph \/_ ps ) <-> ch ) <-> ( ph <-> -. ( ps \/_ ch ) ) )
8 nbbn 
 |-  ( ( -. ( ph \/_ ps ) <-> ch ) <-> -. ( ( ph \/_ ps ) <-> ch ) )
9 1 7 8 3bitr2ri 
 |-  ( -. ( ( ph \/_ ps ) <-> ch ) <-> -. ( ph <-> ( ps \/_ ch ) ) )
10 df-xor 
 |-  ( ( ( ph \/_ ps ) \/_ ch ) <-> -. ( ( ph \/_ ps ) <-> ch ) )
11 df-xor 
 |-  ( ( ph \/_ ( ps \/_ ch ) ) <-> -. ( ph <-> ( ps \/_ ch ) ) )
12 9 10 11 3bitr4i 
 |-  ( ( ( ph \/_ ps ) \/_ ch ) <-> ( ph \/_ ( ps \/_ ch ) ) )