Theorem 3vth5 790

Description: A 3-variable theorem.

Assertion

Ref	Expression
3vth5	((a ->2 b)_|_ ->2 (b v c)) = (c v ((a ->2 b) ^ (c ->2 b)))

Proof of Theorem 3vth5

Step	Hyp	Ref	Expression
1		ax-a3 31	. . 3 ((b v c) v ((b v (a_|_ ^ b_|_)) ^ (b v c)_|_)) = (b v (c v ((b v (a_|_ ^ b_|_)) ^ (b v c)_|_)))
2		or12 73	. . . 4 (b v (c v ((b v (a_|_ ^ b_|_)) ^ (b v c)_|_))) = (c v (b v ((b v (a_|_ ^ b_|_)) ^ (b v c)_|_)))
3		comorr 176	. . . . . . 7 b C (b v (a_|_ ^ b_|_))
4		comorr 176	. . . . . . . 8 b C (b v c)
5	4	comcom2 175	. . . . . . 7 b C (b v c)_|_
6	3, 5	fh3 453	. . . . . 6 (b v ((b v (a_|_ ^ b_|_)) ^ (b v c)_|_)) = ((b v (b v (a_|_ ^ b_|_))) ^ (b v (b v c)_|_))
7		ax-a3 31	. . . . . . . . 9 ((b v b) v (a_|_ ^ b_|_)) = (b v (b v (a_|_ ^ b_|_)))
8	7	ax-r1 34	. . . . . . . 8 (b v (b v (a_|_ ^ b_|_))) = ((b v b) v (a_|_ ^ b_|_))
9		oridm 102	. . . . . . . . 9 (b v b) = b
10	9	ax-r5 37	. . . . . . . 8 ((b v b) v (a_|_ ^ b_|_)) = (b v (a_|_ ^ b_|_))
11	8, 10	ax-r2 35	. . . . . . 7 (b v (b v (a_|_ ^ b_|_))) = (b v (a_|_ ^ b_|_))
12		ancom 68	. . . . . . . . . 10 (c_|_ ^ b_|_) = (b_|_ ^ c_|_)
13		anor3 82	. . . . . . . . . 10 (b_|_ ^ c_|_) = (b v c)_|_
14	12, 13	ax-r2 35	. . . . . . . . 9 (c_|_ ^ b_|_) = (b v c)_|_
15	14	ax-r1 34	. . . . . . . 8 (b v c)_|_ = (c_|_ ^ b_|_)
16	15	lor 66	. . . . . . 7 (b v (b v c)_|_) = (b v (c_|_ ^ b_|_))
17	11, 16	2an 72	. . . . . 6 ((b v (b v (a_|_ ^ b_|_))) ^ (b v (b v c)_|_)) = ((b v (a_|_ ^ b_|_)) ^ (b v (c_|_ ^ b_|_)))
18	6, 17	ax-r2 35	. . . . 5 (b v ((b v (a_|_ ^ b_|_)) ^ (b v c)_|_)) = ((b v (a_|_ ^ b_|_)) ^ (b v (c_|_ ^ b_|_)))
19	18	lor 66	. . . 4 (c v (b v ((b v (a_|_ ^ b_|_)) ^ (b v c)_|_))) = (c v ((b v (a_|_ ^ b_|_)) ^ (b v (c_|_ ^ b_|_))))
20	2, 19	ax-r2 35	. . 3 (b v (c v ((b v (a_|_ ^ b_|_)) ^ (b v c)_|_))) = (c v ((b v (a_|_ ^ b_|_)) ^ (b v (c_|_ ^ b_|_))))
21	1, 20	ax-r2 35	. 2 ((b v c) v ((b v (a_|_ ^ b_|_)) ^ (b v c)_|_)) = (c v ((b v (a_|_ ^ b_|_)) ^ (b v (c_|_ ^ b_|_))))
22		df-i2 44	. . 3 ((a ->2 b)_|_ ->2 (b v c)) = ((b v c) v ((a ->2 b)_|__|_ ^ (b v c)_|_))
23		df-i2 44	. . . . . . . 8 (a ->2 b) = (b v (a_|_ ^ b_|_))
24	23	ax-r1 34	. . . . . . 7 (b v (a_|_ ^ b_|_)) = (a ->2 b)
25		ax-a1 29	. . . . . . 7 (a ->2 b) = (a ->2 b)_|__|_
26	24, 25	ax-r2 35	. . . . . 6 (b v (a_|_ ^ b_|_)) = (a ->2 b)_|__|_
27	26	ran 71	. . . . 5 ((b v (a_|_ ^ b_|_)) ^ (b v c)_|_) = ((a ->2 b)_|__|_ ^ (b v c)_|_)
28	27	lor 66	. . . 4 ((b v c) v ((b v (a_|_ ^ b_|_)) ^ (b v c)_|_)) = ((b v c) v ((a ->2 b)_|__|_ ^ (b v c)_|_))
29	28	ax-r1 34	. . 3 ((b v c) v ((a ->2 b)_|__|_ ^ (b v c)_|_)) = ((b v c) v ((b v (a_|_ ^ b_|_)) ^ (b v c)_|_))
30	22, 29	ax-r2 35	. 2 ((a ->2 b)_|_ ->2 (b v c)) = ((b v c) v ((b v (a_|_ ^ b_|_)) ^ (b v c)_|_))
31		df-i2 44	. . . 4 (c ->2 b) = (b v (c_|_ ^ b_|_))
32	23, 31	2an 72	. . 3 ((a ->2 b) ^ (c ->2 b)) = ((b v (a_|_ ^ b_|_)) ^ (b v (c_|_ ^ b_|_)))
33	32	lor 66	. 2 (c v ((a ->2 b) ^ (c ->2 b))) = (c v ((b v (a_|_ ^ b_|_)) ^ (b v (c_|_ ^ b_|_))))
34	21, 30, 33	3tr1 60	1 ((a ->2 b)_|_ ->2 (b v c)) = (c v ((a ->2 b) ^ (c ->2 b)))

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

This theorem is referenced by:  3vth6 791  3vth7 792

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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