Theorem 3vth9 794

Description: A 3-variable theorem.

Assertion

Ref	Expression
3vth9	((a v b) ->1 (c ->2 b)) = ((b v c) ->2 (a ->2 b))

Proof of Theorem 3vth9

Step	Hyp	Ref	Expression
1		anor3 82	. . . 4 (a_|_ ^ b_|_) = (a v b)_|_
2	1	ax-r1 34	. . 3 (a v b)_|_ = (a_|_ ^ b_|_)
3		df-i2 44	. . . 4 (c ->2 b) = (b v (c_|_ ^ b_|_))
4	3	lan 70	. . 3 ((a v b) ^ (c ->2 b)) = ((a v b) ^ (b v (c_|_ ^ b_|_)))
5	2, 4	2or 67	. 2 ((a v b)_|_ v ((a v b) ^ (c ->2 b))) = ((a_|_ ^ b_|_) v ((a v b) ^ (b v (c_|_ ^ b_|_))))
6		df-i1 43	. 2 ((a v b) ->1 (c ->2 b)) = ((a v b)_|_ v ((a v b) ^ (c ->2 b)))
7		df-i2 44	. . . 4 ((b v c) ->2 (a ->2 b)) = ((a ->2 b) v ((b v c)_|_ ^ (a ->2 b)_|_))
8		df-i2 44	. . . . 5 (a ->2 b) = (b v (a_|_ ^ b_|_))
9		anor3 82	. . . . . . . 8 (b_|_ ^ c_|_) = (b v c)_|_
10	9	ax-r1 34	. . . . . . 7 (b v c)_|_ = (b_|_ ^ c_|_)
11		ud2lem0c 270	. . . . . . 7 (a ->2 b)_|_ = (b_|_ ^ (a v b))
12	10, 11	2an 72	. . . . . 6 ((b v c)_|_ ^ (a ->2 b)_|_) = ((b_|_ ^ c_|_) ^ (b_|_ ^ (a v b)))
13		anandi 106	. . . . . . . 8 (b_|_ ^ (c_|_ ^ (a v b))) = ((b_|_ ^ c_|_) ^ (b_|_ ^ (a v b)))
14	13	ax-r1 34	. . . . . . 7 ((b_|_ ^ c_|_) ^ (b_|_ ^ (a v b))) = (b_|_ ^ (c_|_ ^ (a v b)))
15		anass 69	. . . . . . . 8 ((b_|_ ^ c_|_) ^ (a v b)) = (b_|_ ^ (c_|_ ^ (a v b)))
16	15	ax-r1 34	. . . . . . 7 (b_|_ ^ (c_|_ ^ (a v b))) = ((b_|_ ^ c_|_) ^ (a v b))
17	14, 16	ax-r2 35	. . . . . 6 ((b_|_ ^ c_|_) ^ (b_|_ ^ (a v b))) = ((b_|_ ^ c_|_) ^ (a v b))
18	12, 17	ax-r2 35	. . . . 5 ((b v c)_|_ ^ (a ->2 b)_|_) = ((b_|_ ^ c_|_) ^ (a v b))
19	8, 18	2or 67	. . . 4 ((a ->2 b) v ((b v c)_|_ ^ (a ->2 b)_|_)) = ((b v (a_|_ ^ b_|_)) v ((b_|_ ^ c_|_) ^ (a v b)))
20	7, 19	ax-r2 35	. . 3 ((b v c) ->2 (a ->2 b)) = ((b v (a_|_ ^ b_|_)) v ((b_|_ ^ c_|_) ^ (a v b)))
21		or32 75	. . . 4 ((b v (a_|_ ^ b_|_)) v ((b_|_ ^ c_|_) ^ (a v b))) = ((b v ((b_|_ ^ c_|_) ^ (a v b))) v (a_|_ ^ b_|_))
22		comanr1 446	. . . . . . . . 9 b_|_ C (b_|_ ^ c_|_)
23	22	comcom6 441	. . . . . . . 8 b C (b_|_ ^ c_|_)
24		comorr2 445	. . . . . . . 8 b C (a v b)
25	23, 24	fh3 453	. . . . . . 7 (b v ((b_|_ ^ c_|_) ^ (a v b))) = ((b v (b_|_ ^ c_|_)) ^ (b v (a v b)))
26		ancom 68	. . . . . . . . . 10 (b_|_ ^ c_|_) = (c_|_ ^ b_|_)
27	26	lor 66	. . . . . . . . 9 (b v (b_|_ ^ c_|_)) = (b v (c_|_ ^ b_|_))
28		or12 73	. . . . . . . . . 10 (b v (a v b)) = (a v (b v b))
29		oridm 102	. . . . . . . . . . 11 (b v b) = b
30	29	lor 66	. . . . . . . . . 10 (a v (b v b)) = (a v b)
31	28, 30	ax-r2 35	. . . . . . . . 9 (b v (a v b)) = (a v b)
32	27, 31	2an 72	. . . . . . . 8 ((b v (b_|_ ^ c_|_)) ^ (b v (a v b))) = ((b v (c_|_ ^ b_|_)) ^ (a v b))
33		ancom 68	. . . . . . . 8 ((b v (c_|_ ^ b_|_)) ^ (a v b)) = ((a v b) ^ (b v (c_|_ ^ b_|_)))
34	32, 33	ax-r2 35	. . . . . . 7 ((b v (b_|_ ^ c_|_)) ^ (b v (a v b))) = ((a v b) ^ (b v (c_|_ ^ b_|_)))
35	25, 34	ax-r2 35	. . . . . 6 (b v ((b_|_ ^ c_|_) ^ (a v b))) = ((a v b) ^ (b v (c_|_ ^ b_|_)))
36	35	ax-r5 37	. . . . 5 ((b v ((b_|_ ^ c_|_) ^ (a v b))) v (a_|_ ^ b_|_)) = (((a v b) ^ (b v (c_|_ ^ b_|_))) v (a_|_ ^ b_|_))
37		ax-a2 30	. . . . 5 (((a v b) ^ (b v (c_|_ ^ b_|_))) v (a_|_ ^ b_|_)) = ((a_|_ ^ b_|_) v ((a v b) ^ (b v (c_|_ ^ b_|_))))
38	36, 37	ax-r2 35	. . . 4 ((b v ((b_|_ ^ c_|_) ^ (a v b))) v (a_|_ ^ b_|_)) = ((a_|_ ^ b_|_) v ((a v b) ^ (b v (c_|_ ^ b_|_))))
39	21, 38	ax-r2 35	. . 3 ((b v (a_|_ ^ b_|_)) v ((b_|_ ^ c_|_) ^ (a v b))) = ((a_|_ ^ b_|_) v ((a v b) ^ (b v (c_|_ ^ b_|_))))
40	20, 39	ax-r2 35	. 2 ((b v c) ->2 (a ->2 b)) = ((a_|_ ^ b_|_) v ((a v b) ^ (b v (c_|_ ^ b_|_))))
41	5, 6, 40	3tr1 60	1 ((a v b) ->1 (c ->2 b)) = ((b v c) ->2 (a ->2 b))

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

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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