Theorem axoa4a 1016

Description: Proper 4-variable OA law variant.

Assertion

Ref	Expression
axoa4a	((a ->1 d) ^ (a v (b ^ (((a ^ b) v ((a ->1 d) ^ (b ->1 d))) v (((a ^ c) v ((a ->1 d) ^ (c ->1 d))) ^ ((b ^ c) v ((b ->1 d) ^ (c ->1 d)))))))) =< (((a ^ d) v (b ^ d)) v (c ^ d))

Proof of Theorem axoa4a

Step	Hyp	Ref	Expression
1		id 58	. 2 (a ->1 d)_|_ = (a ->1 d)_|_
2		id 58	. 2 (b ->1 d)_|_ = (b ->1 d)_|_
3		id 58	. 2 (c ->1 d)_|_ = (c ->1 d)_|_
4		leo 150	. . . 4 a_|_ =< (a_|_ v (a ^ d))
5		df-i1 43	. . . . . 6 (a ->1 d) = (a_|_ v (a ^ d))
6	5	ax-r1 34	. . . . 5 (a_|_ v (a ^ d)) = (a ->1 d)
7		ax-a1 29	. . . . 5 (a ->1 d) = (a ->1 d)_|__|_
8	6, 7	ax-r2 35	. . . 4 (a_|_ v (a ^ d)) = (a ->1 d)_|__|_
9	4, 8	lbtr 131	. . 3 a_|_ =< (a ->1 d)_|__|_
10		leo 150	. . . 4 b_|_ =< (b_|_ v (b ^ d))
11		df-i1 43	. . . . . 6 (b ->1 d) = (b_|_ v (b ^ d))
12	11	ax-r1 34	. . . . 5 (b_|_ v (b ^ d)) = (b ->1 d)
13		ax-a1 29	. . . . 5 (b ->1 d) = (b ->1 d)_|__|_
14	12, 13	ax-r2 35	. . . 4 (b_|_ v (b ^ d)) = (b ->1 d)_|__|_
15	10, 14	lbtr 131	. . 3 b_|_ =< (b ->1 d)_|__|_
16		leo 150	. . . 4 c_|_ =< (c_|_ v (c ^ d))
17		df-i1 43	. . . . . 6 (c ->1 d) = (c_|_ v (c ^ d))
18	17	ax-r1 34	. . . . 5 (c_|_ v (c ^ d)) = (c ->1 d)
19		ax-a1 29	. . . . 5 (c ->1 d) = (c ->1 d)_|__|_
20	18, 19	ax-r2 35	. . . 4 (c_|_ v (c ^ d)) = (c ->1 d)_|__|_
21	16, 20	lbtr 131	. . 3 c_|_ =< (c ->1 d)_|__|_
22	9, 15, 21	oa6 1015	. 2 (((a_|_ v (a ->1 d)_|_) ^ (b_|_ v (b ->1 d)_|_)) ^ (c_|_ v (c ->1 d)_|_)) =< ((a ->1 d)_|_ v (a_|_ ^ (b_|_ v (((a_|_ v b_|_) ^ ((a ->1 d)_|_ v (b ->1 d)_|_)) ^ (((a_|_ v c_|_) ^ ((a ->1 d)_|_ v (c ->1 d)_|_)) v ((b_|_ v c_|_) ^ ((b ->1 d)_|_ v (c ->1 d)_|_)))))))
23	1, 2, 3, 22	oa6to4 938	1 ((a ->1 d) ^ (a v (b ^ (((a ^ b) v ((a ->1 d) ^ (b ->1 d))) v (((a ^ c) v ((a ->1 d) ^ (c ->1 d))) ^ ((b ^ c) v ((b ->1 d) ^ (c ->1 d)))))))) =< (((a ^ d) v (b ^ d)) v (c ^ d))

Syntax hints:   =< wle 2  _|_wn 4   v wo 6   ^ wa 7   ->1 wi1 13

This theorem is referenced by:  4oa 1018

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  ax-4oa 1012

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-i1 43  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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