Theorem 2oath1 808

Description: OA-like theorem with ->2 instead of ->0.

Assertion

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

Proof of Theorem 2oath1

Step	Hyp	Ref	Expression
1		df-i2 44	. . 3 ((b v c) ->2 ((a ->2 b) ^ (a ->2 c))) = (((a ->2 b) ^ (a ->2 c)) v ((b v c)_|_ ^ ((a ->2 b) ^ (a ->2 c))_|_))
2	1	lan 70	. 2 ((a ->2 b) ^ ((b v c) ->2 ((a ->2 b) ^ (a ->2 c)))) = ((a ->2 b) ^ (((a ->2 b) ^ (a ->2 c)) v ((b v c)_|_ ^ ((a ->2 b) ^ (a ->2 c))_|_)))
3		coman1 177	. . 3 ((a ->2 b) ^ (a ->2 c)) C (a ->2 b)
4		comorr2 445	. . . . 5 ((a ->2 b) ^ (a ->2 c)) C ((b v c) v ((a ->2 b) ^ (a ->2 c)))
5	4	comcom2 175	. . . 4 ((a ->2 b) ^ (a ->2 c)) C ((b v c) v ((a ->2 b) ^ (a ->2 c)))_|_
6		anor3 82	. . . . 5 ((b v c)_|_ ^ ((a ->2 b) ^ (a ->2 c))_|_) = ((b v c) v ((a ->2 b) ^ (a ->2 c)))_|_
7	6	ax-r1 34	. . . 4 ((b v c) v ((a ->2 b) ^ (a ->2 c)))_|_ = ((b v c)_|_ ^ ((a ->2 b) ^ (a ->2 c))_|_)
8	5, 7	cbtr 174	. . 3 ((a ->2 b) ^ (a ->2 c)) C ((b v c)_|_ ^ ((a ->2 b) ^ (a ->2 c))_|_)
9	3, 8	fh2 452	. 2 ((a ->2 b) ^ (((a ->2 b) ^ (a ->2 c)) v ((b v c)_|_ ^ ((a ->2 b) ^ (a ->2 c))_|_))) = (((a ->2 b) ^ ((a ->2 b) ^ (a ->2 c))) v ((a ->2 b) ^ ((b v c)_|_ ^ ((a ->2 b) ^ (a ->2 c))_|_)))
10		anass 69	. . . . . 6 (((a ->2 b) ^ (a ->2 b)) ^ (a ->2 c)) = ((a ->2 b) ^ ((a ->2 b) ^ (a ->2 c)))
11	10	ax-r1 34	. . . . 5 ((a ->2 b) ^ ((a ->2 b) ^ (a ->2 c))) = (((a ->2 b) ^ (a ->2 b)) ^ (a ->2 c))
12		anidm 103	. . . . . 6 ((a ->2 b) ^ (a ->2 b)) = (a ->2 b)
13	12	ran 71	. . . . 5 (((a ->2 b) ^ (a ->2 b)) ^ (a ->2 c)) = ((a ->2 b) ^ (a ->2 c))
14	11, 13	ax-r2 35	. . . 4 ((a ->2 b) ^ ((a ->2 b) ^ (a ->2 c))) = ((a ->2 b) ^ (a ->2 c))
15		oran 79	. . . . . . . . 9 ((b v c) v ((a ->2 b) ^ (a ->2 c))) = ((b v c)_|_ ^ ((a ->2 b) ^ (a ->2 c))_|_)_|_
16	15	lor 66	. . . . . . . 8 ((a ->2 b)_|_ v ((b v c) v ((a ->2 b) ^ (a ->2 c)))) = ((a ->2 b)_|_ v ((b v c)_|_ ^ ((a ->2 b) ^ (a ->2 c))_|_)_|_)
17	16	ax-r1 34	. . . . . . 7 ((a ->2 b)_|_ v ((b v c)_|_ ^ ((a ->2 b) ^ (a ->2 c))_|_)_|_) = ((a ->2 b)_|_ v ((b v c) v ((a ->2 b) ^ (a ->2 c))))
18		2oalem1 807	. . . . . . 7 ((a ->2 b)_|_ v ((b v c) v ((a ->2 b) ^ (a ->2 c)))) = 1
19	17, 18	ax-r2 35	. . . . . 6 ((a ->2 b)_|_ v ((b v c)_|_ ^ ((a ->2 b) ^ (a ->2 c))_|_)_|_) = 1
20	19	ax-r4 36	. . . . 5 ((a ->2 b)_|_ v ((b v c)_|_ ^ ((a ->2 b) ^ (a ->2 c))_|_)_|_)_|_ = 1_|_
21		df-a 39	. . . . 5 ((a ->2 b) ^ ((b v c)_|_ ^ ((a ->2 b) ^ (a ->2 c))_|_)) = ((a ->2 b)_|_ v ((b v c)_|_ ^ ((a ->2 b) ^ (a ->2 c))_|_)_|_)_|_
22		df-f 41	. . . . 5 0 = 1_|_
23	20, 21, 22	3tr1 60	. . . 4 ((a ->2 b) ^ ((b v c)_|_ ^ ((a ->2 b) ^ (a ->2 c))_|_)) = 0
24	14, 23	2or 67	. . 3 (((a ->2 b) ^ ((a ->2 b) ^ (a ->2 c))) v ((a ->2 b) ^ ((b v c)_|_ ^ ((a ->2 b) ^ (a ->2 c))_|_))) = (((a ->2 b) ^ (a ->2 c)) v 0)
25		or0 94	. . 3 (((a ->2 b) ^ (a ->2 c)) v 0) = ((a ->2 b) ^ (a ->2 c))
26	24, 25	ax-r2 35	. 2 (((a ->2 b) ^ ((a ->2 b) ^ (a ->2 c))) v ((a ->2 b) ^ ((b v c)_|_ ^ ((a ->2 b) ^ (a ->2 c))_|_))) = ((a ->2 b) ^ (a ->2 c))
27	2, 9, 26	3tr 62	1 ((a ->2 b) ^ ((b v c) ->2 ((a ->2 b) ^ (a ->2 c)))) = ((a ->2 b) ^ (a ->2 c))

Syntax hints:   = wb 1  _|_wn 4   v wo 6   ^ wa 7  1wt 9  0wf 10   ->2 wi2 14

This theorem is referenced by:  2oath1i1 809  oale 811  oaeqv 812  distoa 924

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a4 32  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37  ax-r3 421

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-i2 44  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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