Theorem 3vth6 791

Description: A 3-variable theorem.

Assertion

Ref	Expression
3vth6	((a ->2 b)_|_ ->2 (b v c)) = (((a ->2 b) ^ (c ->2 b)) v ((a ->2 c) ^ (b ->2 c)))

Proof of Theorem 3vth6

Step	Hyp	Ref	Expression
1		oridm 102	. . 3 (((a ->2 b)_|_ ->2 (b v c)) v ((a ->2 b)_|_ ->2 (b v c))) = ((a ->2 b)_|_ ->2 (b v c))
2	1	ax-r1 34	. 2 ((a ->2 b)_|_ ->2 (b v c)) = (((a ->2 b)_|_ ->2 (b v c)) v ((a ->2 b)_|_ ->2 (b v c)))
3		3vth4 789	. . . 4 ((a ->2 b)_|_ ->2 (b v c)) = ((a ->2 c)_|_ ->2 (b v c))
4	3	lor 66	. . 3 (((a ->2 b)_|_ ->2 (b v c)) v ((a ->2 b)_|_ ->2 (b v c))) = (((a ->2 b)_|_ ->2 (b v c)) v ((a ->2 c)_|_ ->2 (b v c)))
5		3vth5 790	. . . . 5 ((a ->2 b)_|_ ->2 (b v c)) = (c v ((a ->2 b) ^ (c ->2 b)))
6		ax-a2 30	. . . . . . 7 (b v c) = (c v b)
7	6	ud2lem0a 250	. . . . . 6 ((a ->2 c)_|_ ->2 (b v c)) = ((a ->2 c)_|_ ->2 (c v b))
8		3vth5 790	. . . . . 6 ((a ->2 c)_|_ ->2 (c v b)) = (b v ((a ->2 c) ^ (b ->2 c)))
9	7, 8	ax-r2 35	. . . . 5 ((a ->2 c)_|_ ->2 (b v c)) = (b v ((a ->2 c) ^ (b ->2 c)))
10	5, 9	2or 67	. . . 4 (((a ->2 b)_|_ ->2 (b v c)) v ((a ->2 c)_|_ ->2 (b v c))) = ((c v ((a ->2 b) ^ (c ->2 b))) v (b v ((a ->2 c) ^ (b ->2 c))))
11		or4 77	. . . . 5 ((c v ((a ->2 b) ^ (c ->2 b))) v (b v ((a ->2 c) ^ (b ->2 c)))) = ((c v b) v (((a ->2 b) ^ (c ->2 b)) v ((a ->2 c) ^ (b ->2 c))))
12		ax-a2 30	. . . . . . 7 (c v b) = (b v c)
13	12	ax-r5 37	. . . . . 6 ((c v b) v (((a ->2 b) ^ (c ->2 b)) v ((a ->2 c) ^ (b ->2 c)))) = ((b v c) v (((a ->2 b) ^ (c ->2 b)) v ((a ->2 c) ^ (b ->2 c))))
14		or4 77	. . . . . . 7 ((b v c) v (((a ->2 b) ^ (c ->2 b)) v ((a ->2 c) ^ (b ->2 c)))) = ((b v ((a ->2 b) ^ (c ->2 b))) v (c v ((a ->2 c) ^ (b ->2 c))))
15		leo 150	. . . . . . . . . . 11 b =< (b v (a_|_ ^ b_|_))
16		df-i2 44	. . . . . . . . . . . 12 (a ->2 b) = (b v (a_|_ ^ b_|_))
17	16	ax-r1 34	. . . . . . . . . . 11 (b v (a_|_ ^ b_|_)) = (a ->2 b)
18	15, 17	lbtr 131	. . . . . . . . . 10 b =< (a ->2 b)
19		leo 150	. . . . . . . . . . 11 b =< (b v (c_|_ ^ b_|_))
20		df-i2 44	. . . . . . . . . . . 12 (c ->2 b) = (b v (c_|_ ^ b_|_))
21	20	ax-r1 34	. . . . . . . . . . 11 (b v (c_|_ ^ b_|_)) = (c ->2 b)
22	19, 21	lbtr 131	. . . . . . . . . 10 b =< (c ->2 b)
23	18, 22	ler2an 165	. . . . . . . . 9 b =< ((a ->2 b) ^ (c ->2 b))
24	23	df-le2 123	. . . . . . . 8 (b v ((a ->2 b) ^ (c ->2 b))) = ((a ->2 b) ^ (c ->2 b))
25		leo 150	. . . . . . . . . . 11 c =< (c v (a_|_ ^ c_|_))
26		df-i2 44	. . . . . . . . . . . 12 (a ->2 c) = (c v (a_|_ ^ c_|_))
27	26	ax-r1 34	. . . . . . . . . . 11 (c v (a_|_ ^ c_|_)) = (a ->2 c)
28	25, 27	lbtr 131	. . . . . . . . . 10 c =< (a ->2 c)
29		leo 150	. . . . . . . . . . 11 c =< (c v (b_|_ ^ c_|_))
30		df-i2 44	. . . . . . . . . . . 12 (b ->2 c) = (c v (b_|_ ^ c_|_))
31	30	ax-r1 34	. . . . . . . . . . 11 (c v (b_|_ ^ c_|_)) = (b ->2 c)
32	29, 31	lbtr 131	. . . . . . . . . 10 c =< (b ->2 c)
33	28, 32	ler2an 165	. . . . . . . . 9 c =< ((a ->2 c) ^ (b ->2 c))
34	33	df-le2 123	. . . . . . . 8 (c v ((a ->2 c) ^ (b ->2 c))) = ((a ->2 c) ^ (b ->2 c))
35	24, 34	2or 67	. . . . . . 7 ((b v ((a ->2 b) ^ (c ->2 b))) v (c v ((a ->2 c) ^ (b ->2 c)))) = (((a ->2 b) ^ (c ->2 b)) v ((a ->2 c) ^ (b ->2 c)))
36	14, 35	ax-r2 35	. . . . . 6 ((b v c) v (((a ->2 b) ^ (c ->2 b)) v ((a ->2 c) ^ (b ->2 c)))) = (((a ->2 b) ^ (c ->2 b)) v ((a ->2 c) ^ (b ->2 c)))
37	13, 36	ax-r2 35	. . . . 5 ((c v b) v (((a ->2 b) ^ (c ->2 b)) v ((a ->2 c) ^ (b ->2 c)))) = (((a ->2 b) ^ (c ->2 b)) v ((a ->2 c) ^ (b ->2 c)))
38	11, 37	ax-r2 35	. . . 4 ((c v ((a ->2 b) ^ (c ->2 b))) v (b v ((a ->2 c) ^ (b ->2 c)))) = (((a ->2 b) ^ (c ->2 b)) v ((a ->2 c) ^ (b ->2 c)))
39	10, 38	ax-r2 35	. . 3 (((a ->2 b)_|_ ->2 (b v c)) v ((a ->2 c)_|_ ->2 (b v c))) = (((a ->2 b) ^ (c ->2 b)) v ((a ->2 c) ^ (b ->2 c)))
40	4, 39	ax-r2 35	. 2 (((a ->2 b)_|_ ->2 (b v c)) v ((a ->2 b)_|_ ->2 (b v c))) = (((a ->2 b) ^ (c ->2 b)) v ((a ->2 c) ^ (b ->2 c)))
41	2, 40	ax-r2 35	1 ((a ->2 b)_|_ ->2 (b v c)) = (((a ->2 b) ^ (c ->2 b)) v ((a ->2 c) ^ (b ->2 c)))

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

This theorem is referenced by:  3vth8 793

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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