Theorem oaliii 981

Description: Orthoarguesian law. Godowski/Greechie, Eq. III.

Assertion

Ref	Expression
oaliii	((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))))

Proof of Theorem oaliii

Step	Hyp	Ref	Expression
1		anass 69	. . . . 5 (((a ->2 b) ^ ((b v c)_|_ v ((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))) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))))
2		anidm 103	. . . . . 6 (((b v c)_|_ v ((a ->2 b) ^ (a ->2 c))) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))) = ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))
3	2	lan 70	. . . . 5 ((a ->2 b) ^ (((b v c)_|_ v ((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))))
4	1, 3	ax-r2 35	. . . 4 (((a ->2 b) ^ ((b v c)_|_ v ((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))))
5	4	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)))) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c))))
6		oal2 979	. . . 4 ((a ->2 b) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))) =< (a ->2 c)
7	6	leran 145	. . 3 (((a ->2 b) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))) =< ((a ->2 c) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c))))
8	5, 7	bltr 130	. 2 ((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))))
9		anass 69	. . . . 5 (((a ->2 c) ^ ((c v b)_|_ v ((a ->2 c) ^ (a ->2 b)))) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))) = ((a ->2 c) ^ (((c v b)_|_ v ((a ->2 c) ^ (a ->2 b))) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))))
10		ax-a2 30	. . . . . . . . . 10 (c v b) = (b v c)
11	10	ax-r4 36	. . . . . . . . 9 (c v b)_|_ = (b v c)_|_
12		ancom 68	. . . . . . . . 9 ((a ->2 c) ^ (a ->2 b)) = ((a ->2 b) ^ (a ->2 c))
13	11, 12	2or 67	. . . . . . . 8 ((c v b)_|_ v ((a ->2 c) ^ (a ->2 b))) = ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))
14	13	ran 71	. . . . . . 7 (((c v b)_|_ v ((a ->2 c) ^ (a ->2 b))) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))) = (((b v c)_|_ v ((a ->2 b) ^ (a ->2 c))) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c))))
15	14, 2	ax-r2 35	. . . . . 6 (((c v b)_|_ v ((a ->2 c) ^ (a ->2 b))) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))) = ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))
16	15	lan 70	. . . . 5 ((a ->2 c) ^ (((c v b)_|_ v ((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))))
17	9, 16	ax-r2 35	. . . 4 (((a ->2 c) ^ ((c v b)_|_ v ((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))))
18	17	ax-r1 34	. . 3 ((a ->2 c) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))) = (((a ->2 c) ^ ((c v b)_|_ v ((a ->2 c) ^ (a ->2 b)))) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c))))
19		oal2 979	. . . 4 ((a ->2 c) ^ ((c v b)_|_ v ((a ->2 c) ^ (a ->2 b)))) =< (a ->2 b)
20	19	leran 145	. . 3 (((a ->2 c) ^ ((c v b)_|_ v ((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))))
21	18, 20	bltr 130	. 2 ((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))))
22	8, 21	lebi 137	1 ((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))))

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

This theorem is referenced by:  oalii 982  oath1 984

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

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