Theorem oath1 984

Description: OA theorem.

Assertion

Ref	Expression
oath1	((a ->2 b) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))) = ((a ->2 b) ^ (a ->2 c))

Proof of Theorem oath1

Step	Hyp	Ref	Expression
1		oaliii 981	. . . 4 ((a ->2 b) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))) = ((a ->2 c) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c))))
2	1	lan 70	. . 3 (((a ->2 b) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))) ^ ((a ->2 b) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c))))) = (((a ->2 b) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))) ^ ((a ->2 c) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))))
3		anidm 103	. . . 4 (((a ->2 b) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))) ^ ((a ->2 b) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c))))) = ((a ->2 b) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c))))
4	3	ax-r1 34	. . 3 ((a ->2 b) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))) = (((a ->2 b) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))) ^ ((a ->2 b) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))))
5		anandir 107	. . 3 (((a ->2 b) ^ (a ->2 c)) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))) = (((a ->2 b) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))) ^ ((a ->2 c) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))))
6	2, 4, 5	3tr1 60	. 2 ((a ->2 b) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))) = (((a ->2 b) ^ (a ->2 c)) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c))))
7		ax-a2 30	. . 3 ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c))) = (((a ->2 b) ^ (a ->2 c)) v (b v c)_|_)
8	7	lan 70	. 2 (((a ->2 b) ^ (a ->2 c)) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))) = (((a ->2 b) ^ (a ->2 c)) ^ (((a ->2 b) ^ (a ->2 c)) v (b v c)_|_))
9		a5c 113	. 2 (((a ->2 b) ^ (a ->2 c)) ^ (((a ->2 b) ^ (a ->2 c)) v (b v c)_|_)) = ((a ->2 b) ^ (a ->2 c))
10	6, 8, 9	3tr 62	1 ((a ->2 b) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))) = ((a ->2 b) ^ (a ->2 c))

Syntax hints:   = wb 1  _|_wn 4   v wo 6   ^ wa 7   ->2 wi2 14

This theorem is referenced by:  oalem2 986  oadist2a 987  oacom 991  oacom3 993  oagen1 994

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37  ax-3oa 978

This theorem depends on definitions:  df-a 39  df-t 40  df-f 41  df-i1 43  df-i2 44  df-le1 122  df-le2 123

metamath.org