Theorem test2 785

Description: Part of an attempt to crack a potential Kalmbach axiom.

Assertion

Ref	Expression
test2	(a v b) =< ((a == b)_|_ v ((c v (a ^ b)) ^ (c_|_ v (a ^ b))))

Proof of Theorem test2

Step	Hyp	Ref	Expression
1		dfnb 87	. . . . 5 (a == b)_|_ = ((a v b) ^ (a_|_ v b_|_))
2		anidm 103	. . . . 5 ((a ^ b) ^ (a ^ b)) = (a ^ b)
3	1, 2	2or 67	. . . 4 ((a == b)_|_ v ((a ^ b) ^ (a ^ b))) = (((a v b) ^ (a_|_ v b_|_)) v (a ^ b))
4		comor1 443	. . . . . . 7 (a v b) C a
5		comor2 444	. . . . . . 7 (a v b) C b
6	4, 5	com2an 466	. . . . . 6 (a v b) C (a ^ b)
7	4	comcom2 175	. . . . . . 7 (a v b) C a_|_
8	5	comcom2 175	. . . . . . 7 (a v b) C b_|_
9	7, 8	com2or 465	. . . . . 6 (a v b) C (a_|_ v b_|_)
10	6, 9	fh4r 458	. . . . 5 (((a v b) ^ (a_|_ v b_|_)) v (a ^ b)) = (((a v b) v (a ^ b)) ^ ((a_|_ v b_|_) v (a ^ b)))
11		ax-a2 30	. . . . . . . 8 ((a v b) v (a ^ b)) = ((a ^ b) v (a v b))
12		lea 152	. . . . . . . . . 10 (a ^ b) =< a
13		leo 150	. . . . . . . . . 10 a =< (a v b)
14	12, 13	letr 129	. . . . . . . . 9 (a ^ b) =< (a v b)
15	14	df-le2 123	. . . . . . . 8 ((a ^ b) v (a v b)) = (a v b)
16	11, 15	ax-r2 35	. . . . . . 7 ((a v b) v (a ^ b)) = (a v b)
17		df-a 39	. . . . . . . . 9 (a ^ b) = (a_|_ v b_|_)_|_
18	17	lor 66	. . . . . . . 8 ((a_|_ v b_|_) v (a ^ b)) = ((a_|_ v b_|_) v (a_|_ v b_|_)_|_)
19		df-t 40	. . . . . . . . 9 1 = ((a_|_ v b_|_) v (a_|_ v b_|_)_|_)
20	19	ax-r1 34	. . . . . . . 8 ((a_|_ v b_|_) v (a_|_ v b_|_)_|_) = 1
21	18, 20	ax-r2 35	. . . . . . 7 ((a_|_ v b_|_) v (a ^ b)) = 1
22	16, 21	2an 72	. . . . . 6 (((a v b) v (a ^ b)) ^ ((a_|_ v b_|_) v (a ^ b))) = ((a v b) ^ 1)
23		an1 98	. . . . . 6 ((a v b) ^ 1) = (a v b)
24	22, 23	ax-r2 35	. . . . 5 (((a v b) v (a ^ b)) ^ ((a_|_ v b_|_) v (a ^ b))) = (a v b)
25	10, 24	ax-r2 35	. . . 4 (((a v b) ^ (a_|_ v b_|_)) v (a ^ b)) = (a v b)
26	3, 25	ax-r2 35	. . 3 ((a == b)_|_ v ((a ^ b) ^ (a ^ b))) = (a v b)
27	26	ax-r1 34	. 2 (a v b) = ((a == b)_|_ v ((a ^ b) ^ (a ^ b)))
28		leor 151	. . . 4 (a ^ b) =< (c v (a ^ b))
29		leor 151	. . . 4 (a ^ b) =< (c_|_ v (a ^ b))
30	28, 29	le2an 161	. . 3 ((a ^ b) ^ (a ^ b)) =< ((c v (a ^ b)) ^ (c_|_ v (a ^ b)))
31	30	lelor 158	. 2 ((a == b)_|_ v ((a ^ b) ^ (a ^ b))) =< ((a == b)_|_ v ((c v (a ^ b)) ^ (c_|_ v (a ^ b))))
32	27, 31	bltr 130	1 (a v b) =< ((a == b)_|_ v ((c v (a ^ b)) ^ (c_|_ v (a ^ b))))

Syntax hints:   =< wle 2  _|_wn 4   == tb 5   v wo 6   ^ wa 7  1wt 9

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-le1 122  df-le2 123  df-c1 124  df-c2 125

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