Theorem 3vth7 792

Description: A 3-variable theorem.

Assertion

Ref	Expression
3vth7	((a ->2 b)_|_ ->2 (b v c)) = (a ->2 (b v c))

Proof of Theorem 3vth7

Step	Hyp	Ref	Expression
1		df-i2 44	. . . . 5 (a ->2 b) = (b v (a_|_ ^ b_|_))
2		df-i2 44	. . . . 5 (c ->2 b) = (b v (c_|_ ^ b_|_))
3	1, 2	2an 72	. . . 4 ((a ->2 b) ^ (c ->2 b)) = ((b v (a_|_ ^ b_|_)) ^ (b v (c_|_ ^ b_|_)))
4		anass 69	. . . . . . . . . 10 ((a_|_ ^ b_|_) ^ c_|_) = (a_|_ ^ (b_|_ ^ c_|_))
5	4	ax-r1 34	. . . . . . . . 9 (a_|_ ^ (b_|_ ^ c_|_)) = ((a_|_ ^ b_|_) ^ c_|_)
6		anor3 82	. . . . . . . . . 10 (b_|_ ^ c_|_) = (b v c)_|_
7	6	lan 70	. . . . . . . . 9 (a_|_ ^ (b_|_ ^ c_|_)) = (a_|_ ^ (b v c)_|_)
8		an32 76	. . . . . . . . 9 ((a_|_ ^ b_|_) ^ c_|_) = ((a_|_ ^ c_|_) ^ b_|_)
9	5, 7, 8	3tr2 61	. . . . . . . 8 (a_|_ ^ (b v c)_|_) = ((a_|_ ^ c_|_) ^ b_|_)
10		anidm 103	. . . . . . . . . 10 (b_|_ ^ b_|_) = b_|_
11	10	lan 70	. . . . . . . . 9 ((a_|_ ^ c_|_) ^ (b_|_ ^ b_|_)) = ((a_|_ ^ c_|_) ^ b_|_)
12	11	ax-r1 34	. . . . . . . 8 ((a_|_ ^ c_|_) ^ b_|_) = ((a_|_ ^ c_|_) ^ (b_|_ ^ b_|_))
13		an4 78	. . . . . . . 8 ((a_|_ ^ c_|_) ^ (b_|_ ^ b_|_)) = ((a_|_ ^ b_|_) ^ (c_|_ ^ b_|_))
14	9, 12, 13	3tr 62	. . . . . . 7 (a_|_ ^ (b v c)_|_) = ((a_|_ ^ b_|_) ^ (c_|_ ^ b_|_))
15	14	lor 66	. . . . . 6 (b v (a_|_ ^ (b v c)_|_)) = (b v ((a_|_ ^ b_|_) ^ (c_|_ ^ b_|_)))
16		comanr2 447	. . . . . . . 8 b_|_ C (a_|_ ^ b_|_)
17	16	comcom6 441	. . . . . . 7 b C (a_|_ ^ b_|_)
18		comanr2 447	. . . . . . . 8 b_|_ C (c_|_ ^ b_|_)
19	18	comcom6 441	. . . . . . 7 b C (c_|_ ^ b_|_)
20	17, 19	fh3 453	. . . . . 6 (b v ((a_|_ ^ b_|_) ^ (c_|_ ^ b_|_))) = ((b v (a_|_ ^ b_|_)) ^ (b v (c_|_ ^ b_|_)))
21	15, 20	ax-r2 35	. . . . 5 (b v (a_|_ ^ (b v c)_|_)) = ((b v (a_|_ ^ b_|_)) ^ (b v (c_|_ ^ b_|_)))
22	21	ax-r1 34	. . . 4 ((b v (a_|_ ^ b_|_)) ^ (b v (c_|_ ^ b_|_))) = (b v (a_|_ ^ (b v c)_|_))
23	3, 22	ax-r2 35	. . 3 ((a ->2 b) ^ (c ->2 b)) = (b v (a_|_ ^ (b v c)_|_))
24	23	lor 66	. 2 (c v ((a ->2 b) ^ (c ->2 b))) = (c v (b v (a_|_ ^ (b v c)_|_)))
25		3vth5 790	. 2 ((a ->2 b)_|_ ->2 (b v c)) = (c v ((a ->2 b) ^ (c ->2 b)))
26		df-i2 44	. . 3 (a ->2 (b v c)) = ((b v c) v (a_|_ ^ (b v c)_|_))
27		ax-a3 31	. . 3 ((b v c) v (a_|_ ^ (b v c)_|_)) = (b v (c v (a_|_ ^ (b v c)_|_)))
28		or12 73	. . 3 (b v (c v (a_|_ ^ (b v c)_|_))) = (c v (b v (a_|_ ^ (b v c)_|_)))
29	26, 27, 28	3tr 62	. 2 (a ->2 (b v c)) = (c v (b v (a_|_ ^ (b v c)_|_)))
30	24, 25, 29	3tr1 60	1 ((a ->2 b)_|_ ->2 (b v c)) = (a ->2 (b v 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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