Theorem u4lemoa 605

Description: Lemma for non-tollens implication study.

Assertion

Ref	Expression
u4lemoa	((a ->4 b) v a) = 1

Proof of Theorem u4lemoa

Step	Hyp	Ref	Expression
1		df-i4 46	. . 3 (a ->4 b) = (((a ^ b) v (a_|_ ^ b)) v ((a_|_ v b) ^ b_|_))
2	1	ax-r5 37	. 2 ((a ->4 b) v a) = ((((a ^ b) v (a_|_ ^ b)) v ((a_|_ v b) ^ b_|_)) v a)
3		ax-a3 31	. . 3 ((((a ^ b) v (a_|_ ^ b)) v ((a_|_ v b) ^ b_|_)) v a) = (((a ^ b) v (a_|_ ^ b)) v (((a_|_ v b) ^ b_|_) v a))
4		comor1 443	. . . . . . . 8 (a_|_ v b) C a_|_
5	4	comcom7 442	. . . . . . 7 (a_|_ v b) C a
6		comor2 444	. . . . . . . 8 (a_|_ v b) C b
7	6	comcom2 175	. . . . . . 7 (a_|_ v b) C b_|_
8	5, 7	fh4r 458	. . . . . 6 (((a_|_ v b) ^ b_|_) v a) = (((a_|_ v b) v a) ^ (b_|_ v a))
9		or32 75	. . . . . . . . 9 ((a_|_ v b) v a) = ((a_|_ v a) v b)
10		ax-a2 30	. . . . . . . . . 10 ((a_|_ v a) v b) = (b v (a_|_ v a))
11		df-t 40	. . . . . . . . . . . . . 14 1 = (a v a_|_)
12		ax-a2 30	. . . . . . . . . . . . . 14 (a v a_|_) = (a_|_ v a)
13	11, 12	ax-r2 35	. . . . . . . . . . . . 13 1 = (a_|_ v a)
14	13	lor 66	. . . . . . . . . . . 12 (b v 1) = (b v (a_|_ v a))
15	14	ax-r1 34	. . . . . . . . . . 11 (b v (a_|_ v a)) = (b v 1)
16		or1 96	. . . . . . . . . . 11 (b v 1) = 1
17	15, 16	ax-r2 35	. . . . . . . . . 10 (b v (a_|_ v a)) = 1
18	10, 17	ax-r2 35	. . . . . . . . 9 ((a_|_ v a) v b) = 1
19	9, 18	ax-r2 35	. . . . . . . 8 ((a_|_ v b) v a) = 1
20	19	ran 71	. . . . . . 7 (((a_|_ v b) v a) ^ (b_|_ v a)) = (1 ^ (b_|_ v a))
21		ancom 68	. . . . . . . 8 (1 ^ (b_|_ v a)) = ((b_|_ v a) ^ 1)
22		an1 98	. . . . . . . 8 ((b_|_ v a) ^ 1) = (b_|_ v a)
23	21, 22	ax-r2 35	. . . . . . 7 (1 ^ (b_|_ v a)) = (b_|_ v a)
24	20, 23	ax-r2 35	. . . . . 6 (((a_|_ v b) v a) ^ (b_|_ v a)) = (b_|_ v a)
25	8, 24	ax-r2 35	. . . . 5 (((a_|_ v b) ^ b_|_) v a) = (b_|_ v a)
26	25	lor 66	. . . 4 (((a ^ b) v (a_|_ ^ b)) v (((a_|_ v b) ^ b_|_) v a)) = (((a ^ b) v (a_|_ ^ b)) v (b_|_ v a))
27		ax-a3 31	. . . . 5 (((a ^ b) v (a_|_ ^ b)) v (b_|_ v a)) = ((a ^ b) v ((a_|_ ^ b) v (b_|_ v a)))
28		ax-a2 30	. . . . . . . 8 ((a_|_ ^ b) v (b_|_ v a)) = ((b_|_ v a) v (a_|_ ^ b))
29		ancom 68	. . . . . . . . . . 11 (a_|_ ^ b) = (b ^ a_|_)
30		anor1 80	. . . . . . . . . . 11 (b ^ a_|_) = (b_|_ v a)_|_
31	29, 30	ax-r2 35	. . . . . . . . . 10 (a_|_ ^ b) = (b_|_ v a)_|_
32	31	lor 66	. . . . . . . . 9 ((b_|_ v a) v (a_|_ ^ b)) = ((b_|_ v a) v (b_|_ v a)_|_)
33		df-t 40	. . . . . . . . . 10 1 = ((b_|_ v a) v (b_|_ v a)_|_)
34	33	ax-r1 34	. . . . . . . . 9 ((b_|_ v a) v (b_|_ v a)_|_) = 1
35	32, 34	ax-r2 35	. . . . . . . 8 ((b_|_ v a) v (a_|_ ^ b)) = 1
36	28, 35	ax-r2 35	. . . . . . 7 ((a_|_ ^ b) v (b_|_ v a)) = 1
37	36	lor 66	. . . . . 6 ((a ^ b) v ((a_|_ ^ b) v (b_|_ v a))) = ((a ^ b) v 1)
38		or1 96	. . . . . 6 ((a ^ b) v 1) = 1
39	37, 38	ax-r2 35	. . . . 5 ((a ^ b) v ((a_|_ ^ b) v (b_|_ v a))) = 1
40	27, 39	ax-r2 35	. . . 4 (((a ^ b) v (a_|_ ^ b)) v (b_|_ v a)) = 1
41	26, 40	ax-r2 35	. . 3 (((a ^ b) v (a_|_ ^ b)) v (((a_|_ v b) ^ b_|_) v a)) = 1
42	3, 41	ax-r2 35	. 2 ((((a ^ b) v (a_|_ ^ b)) v ((a_|_ v b) ^ b_|_)) v a) = 1
43	2, 42	ax-r2 35	1 ((a ->4 b) v a) = 1

Syntax hints:   = wb 1  _|_wn 4   v wo 6   ^ wa 7  1wt 9   ->4 wi4 16

This theorem is referenced by:  u4lemnana 630

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-i4 46  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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