Theorem ud4lem2 582

Description: Lemma for unified disjunction.

Assertion

Ref	Expression
ud4lem2	((a v (a' ^ b')) ->4 a) = (a v b)

Proof of Theorem ud4lem2

Step	Hyp	Ref	Expression
1		df-i4 47	. 2 ((a v (a' ^ b')) ->4 a) = ((((a v (a' ^ b')) ^ a) v ((a v (a' ^ b'))' ^ a)) v (((a v (a' ^ b'))' v a) ^ a'))
2		ancom 74	. . . . . . 7 ((a v (a' ^ b')) ^ a) = (a ^ (a v (a' ^ b')))
3		a5c 121	. . . . . . 7 (a ^ (a v (a' ^ b'))) = a
4	2, 3	ax-r2 36	. . . . . 6 ((a v (a' ^ b')) ^ a) = a
5		oran 87	. . . . . . . . 9 (a v (a' ^ b')) = (a' ^ (a' ^ b')')'
6	5	con2 67	. . . . . . . 8 (a v (a' ^ b'))' = (a' ^ (a' ^ b')')
7	6	ran 78	. . . . . . 7 ((a v (a' ^ b'))' ^ a) = ((a' ^ (a' ^ b')') ^ a)
8		ancom 74	. . . . . . . 8 ((a' ^ (a' ^ b')') ^ a) = (a ^ (a' ^ (a' ^ b')'))
9		anass 76	. . . . . . . . . 10 ((a ^ a') ^ (a' ^ b')') = (a ^ (a' ^ (a' ^ b')'))
10	9	ax-r1 35	. . . . . . . . 9 (a ^ (a' ^ (a' ^ b')')) = ((a ^ a') ^ (a' ^ b')')
11		ancom 74	. . . . . . . . . 10 ((a ^ a') ^ (a' ^ b')') = ((a' ^ b')' ^ (a ^ a'))
12		dff 101	. . . . . . . . . . . . 13 0 = (a ^ a')
13	12	lan 77	. . . . . . . . . . . 12 ((a' ^ b')' ^ 0) = ((a' ^ b')' ^ (a ^ a'))
14	13	ax-r1 35	. . . . . . . . . . 11 ((a' ^ b')' ^ (a ^ a')) = ((a' ^ b')' ^ 0)
15		an0 108	. . . . . . . . . . 11 ((a' ^ b')' ^ 0) = 0
16	14, 15	ax-r2 36	. . . . . . . . . 10 ((a' ^ b')' ^ (a ^ a')) = 0
17	11, 16	ax-r2 36	. . . . . . . . 9 ((a ^ a') ^ (a' ^ b')') = 0
18	10, 17	ax-r2 36	. . . . . . . 8 (a ^ (a' ^ (a' ^ b')')) = 0
19	8, 18	ax-r2 36	. . . . . . 7 ((a' ^ (a' ^ b')') ^ a) = 0
20	7, 19	ax-r2 36	. . . . . 6 ((a v (a' ^ b'))' ^ a) = 0
21	4, 20	2or 72	. . . . 5 (((a v (a' ^ b')) ^ a) v ((a v (a' ^ b'))' ^ a)) = (a v 0)
22		or0 102	. . . . 5 (a v 0) = a
23	21, 22	ax-r2 36	. . . 4 (((a v (a' ^ b')) ^ a) v ((a v (a' ^ b'))' ^ a)) = a
24		ancom 74	. . . . 5 (((a v (a' ^ b'))' v a) ^ a') = (a' ^ ((a v (a' ^ b'))' v a))
25		oran 87	. . . . . . . . . . . . 13 (a v b) = (a' ^ b')'
26	25	ax-r1 35	. . . . . . . . . . . 12 (a' ^ b')' = (a v b)
27	26	con3 68	. . . . . . . . . . 11 (a' ^ b') = (a v b)'
28	27	lor 70	. . . . . . . . . 10 (a v (a' ^ b')) = (a v (a v b)')
29		anor2 89	. . . . . . . . . . . 12 (a' ^ (a v b)) = (a v (a v b)')'
30	29	ax-r1 35	. . . . . . . . . . 11 (a v (a v b)')' = (a' ^ (a v b))
31	30	con3 68	. . . . . . . . . 10 (a v (a v b)') = (a' ^ (a v b))'
32	28, 31	ax-r2 36	. . . . . . . . 9 (a v (a' ^ b')) = (a' ^ (a v b))'
33	32	con2 67	. . . . . . . 8 (a v (a' ^ b'))' = (a' ^ (a v b))
34	33	ax-r5 38	. . . . . . 7 ((a v (a' ^ b'))' v a) = ((a' ^ (a v b)) v a)
35		comid 187	. . . . . . . . . 10 a C a
36	35	comcom2 183	. . . . . . . . 9 a C a'
37		comorr 184	. . . . . . . . 9 a C (a v b)
38	36, 37	fh3r 475	. . . . . . . 8 ((a' ^ (a v b)) v a) = ((a' v a) ^ ((a v b) v a))
39		ancom 74	. . . . . . . . . 10 ((a' v a) ^ ((a v b) v a)) = (((a v b) v a) ^ (a' v a))
40		or32 82	. . . . . . . . . . . 12 ((a v b) v a) = ((a v a) v b)
41		oridm 110	. . . . . . . . . . . . 13 (a v a) = a
42	41	ax-r5 38	. . . . . . . . . . . 12 ((a v a) v b) = (a v b)
43	40, 42	ax-r2 36	. . . . . . . . . . 11 ((a v b) v a) = (a v b)
44		df-t 41	. . . . . . . . . . . . 13 1 = (a v a')
45		ax-a2 31	. . . . . . . . . . . . 13 (a v a') = (a' v a)
46	44, 45	ax-r2 36	. . . . . . . . . . . 12 1 = (a' v a)
47	46	ax-r1 35	. . . . . . . . . . 11 (a' v a) = 1
48	43, 47	2an 79	. . . . . . . . . 10 (((a v b) v a) ^ (a' v a)) = ((a v b) ^ 1)
49	39, 48	ax-r2 36	. . . . . . . . 9 ((a' v a) ^ ((a v b) v a)) = ((a v b) ^ 1)
50		an1 106	. . . . . . . . 9 ((a v b) ^ 1) = (a v b)
51	49, 50	ax-r2 36	. . . . . . . 8 ((a' v a) ^ ((a v b) v a)) = (a v b)
52	38, 51	ax-r2 36	. . . . . . 7 ((a' ^ (a v b)) v a) = (a v b)
53	34, 52	ax-r2 36	. . . . . 6 ((a v (a' ^ b'))' v a) = (a v b)
54	53	lan 77	. . . . 5 (a' ^ ((a v (a' ^ b'))' v a)) = (a' ^ (a v b))
55	24, 54	ax-r2 36	. . . 4 (((a v (a' ^ b'))' v a) ^ a') = (a' ^ (a v b))
56	23, 55	2or 72	. . 3 ((((a v (a' ^ b')) ^ a) v ((a v (a' ^ b'))' ^ a)) v (((a v (a' ^ b'))' v a) ^ a')) = (a v (a' ^ (a v b)))
57		oml 445	. . 3 (a v (a' ^ (a v b))) = (a v b)
58	56, 57	ax-r2 36	. 2 ((((a v (a' ^ b')) ^ a) v ((a v (a' ^ b'))' ^ a)) v (((a v (a' ^ b'))' v a) ^ a')) = (a v b)
59	1, 58	ax-r2 36	1 ((a v (a' ^ b')) ->4 a) = (a v b)

Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7  1wt 8  0wf 9   ->4 wi4 15

This theorem is referenced by:  ud4 598

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-r3 439

This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i4 47  df-le1 130  df-le2 131  df-c1 132  df-c2 133

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