Theorem mloa 998

Description: Mladen's OA

Assertion

Ref	Expression
mloa	((a == b) ^ ((b == c) v ((b v (a == b)) ^ (c v (a == c))))) =< (c v (a == c))

Proof of Theorem mloa

Step	Hyp	Ref	Expression
1		lea 152	. . . 4 ((a ->2 b) ^ (b ->2 a)) =< (a ->2 b)
2		ax-a3 31	. . . . . 6 (((b ^ c) v (b_|_ ^ c_|_)) v ((a ->2 b) ^ (a ->2 c))) = ((b ^ c) v ((b_|_ ^ c_|_) v ((a ->2 b) ^ (a ->2 c))))
3		or12 73	. . . . . . 7 ((b ^ c) v ((b_|_ ^ c_|_) v ((a ->2 b) ^ (a ->2 c)))) = ((b_|_ ^ c_|_) v ((b ^ c) v ((a ->2 b) ^ (a ->2 c))))
4		anor3 82	. . . . . . . 8 (b_|_ ^ c_|_) = (b v c)_|_
5	4	ax-r5 37	. . . . . . 7 ((b_|_ ^ c_|_) v ((b ^ c) v ((a ->2 b) ^ (a ->2 c)))) = ((b v c)_|_ v ((b ^ c) v ((a ->2 b) ^ (a ->2 c))))
6	3, 5	ax-r2 35	. . . . . 6 ((b ^ c) v ((b_|_ ^ c_|_) v ((a ->2 b) ^ (a ->2 c)))) = ((b v c)_|_ v ((b ^ c) v ((a ->2 b) ^ (a ->2 c))))
7	2, 6	ax-r2 35	. . . . 5 (((b ^ c) v (b_|_ ^ c_|_)) v ((a ->2 b) ^ (a ->2 c))) = ((b v c)_|_ v ((b ^ c) v ((a ->2 b) ^ (a ->2 c))))
8		leo 150	. . . . . . . . 9 b =< (b v (a_|_ ^ b_|_))
9		df-i2 44	. . . . . . . . . 10 (a ->2 b) = (b v (a_|_ ^ b_|_))
10	9	ax-r1 34	. . . . . . . . 9 (b v (a_|_ ^ b_|_)) = (a ->2 b)
11	8, 10	lbtr 131	. . . . . . . 8 b =< (a ->2 b)
12		leo 150	. . . . . . . . 9 c =< (c v (a_|_ ^ c_|_))
13		df-i2 44	. . . . . . . . . 10 (a ->2 c) = (c v (a_|_ ^ c_|_))
14	13	ax-r1 34	. . . . . . . . 9 (c v (a_|_ ^ c_|_)) = (a ->2 c)
15	12, 14	lbtr 131	. . . . . . . 8 c =< (a ->2 c)
16	11, 15	le2an 161	. . . . . . 7 (b ^ c) =< ((a ->2 b) ^ (a ->2 c))
17		id 58	. . . . . . . 8 ((a ->2 b) ^ (a ->2 c)) = ((a ->2 b) ^ (a ->2 c))
18	17	bile 134	. . . . . . 7 ((a ->2 b) ^ (a ->2 c)) =< ((a ->2 b) ^ (a ->2 c))
19	16, 18	lel2or 162	. . . . . 6 ((b ^ c) v ((a ->2 b) ^ (a ->2 c))) =< ((a ->2 b) ^ (a ->2 c))
20	19	lelor 158	. . . . 5 ((b v c)_|_ v ((b ^ c) v ((a ->2 b) ^ (a ->2 c)))) =< ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))
21	7, 20	bltr 130	. . . 4 (((b ^ c) v (b_|_ ^ c_|_)) v ((a ->2 b) ^ (a ->2 c))) =< ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))
22	1, 21	le2an 161	. . 3 (((a ->2 b) ^ (b ->2 a)) ^ (((b ^ c) v (b_|_ ^ c_|_)) v ((a ->2 b) ^ (a ->2 c)))) =< ((a ->2 b) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c))))
23		oal2 979	. . 3 ((a ->2 b) ^ ((b v c)_|_ v ((a ->2 b) ^ (a ->2 c)))) =< (a ->2 c)
24	22, 23	letr 129	. 2 (((a ->2 b) ^ (b ->2 a)) ^ (((b ^ c) v (b_|_ ^ c_|_)) v ((a ->2 b) ^ (a ->2 c)))) =< (a ->2 c)
25		u2lembi 703	. . 3 ((a ->2 b) ^ (b ->2 a)) = (a == b)
26		dfb 86	. . . . 5 (b == c) = ((b ^ c) v (b_|_ ^ c_|_))
27	26	ax-r1 34	. . . 4 ((b ^ c) v (b_|_ ^ c_|_)) = (b == c)
28		i2bi 704	. . . . 5 (a ->2 b) = (b v (a == b))
29		i2bi 704	. . . . 5 (a ->2 c) = (c v (a == c))
30	28, 29	2an 72	. . . 4 ((a ->2 b) ^ (a ->2 c)) = ((b v (a == b)) ^ (c v (a == c)))
31	27, 30	2or 67	. . 3 (((b ^ c) v (b_|_ ^ c_|_)) v ((a ->2 b) ^ (a ->2 c))) = ((b == c) v ((b v (a == b)) ^ (c v (a == c))))
32	25, 31	2an 72	. 2 (((a ->2 b) ^ (b ->2 a)) ^ (((b ^ c) v (b_|_ ^ c_|_)) v ((a ->2 b) ^ (a ->2 c)))) = ((a == b) ^ ((b == c) v ((b v (a == b)) ^ (c v (a == c)))))
33	24, 32, 29	le3tr2 133	1 ((a == b) ^ ((b == c) v ((b v (a == b)) ^ (c v (a == c))))) =< (c v (a == c))

Syntax hints:   =< wle 2  _|_wn 4   == tb 5   v wo 6   ^ wa 7   ->2 wi2 14

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-3oa 978

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-i1 43  df-i2 44  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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