Theorem wwfh2 217

Description: Foulis-Holland Theorem (weak).

Hypotheses

Ref	Expression
wwfh2.1	a C b
wwfh2.2	c' C a

Assertion

Ref	Expression
wwfh2	((b ^ (a v c)) == ((b ^ a) v (b ^ c))) = 1

Proof of Theorem wwfh2

Step	Hyp	Ref	Expression
1		bicom 96	. 2 ((b ^ (a v c)) == ((b ^ a) v (b ^ c))) = (((b ^ a) v (b ^ c)) == (b ^ (a v c)))
2		ledi 174	. . 3 ((b ^ a) v (b ^ c)) =< (b ^ (a v c))
3		oran 87	. . . . . . . . . . 11 ((b ^ a) v (b ^ c)) = ((b ^ a)' ^ (b ^ c)')'
4		df-a 40	. . . . . . . . . . . . . 14 (b ^ a) = (b' v a')'
5	4	con2 67	. . . . . . . . . . . . 13 (b ^ a)' = (b' v a')
6	5	ran 78	. . . . . . . . . . . 12 ((b ^ a)' ^ (b ^ c)') = ((b' v a') ^ (b ^ c)')
7	6	ax-r4 37	. . . . . . . . . . 11 ((b ^ a)' ^ (b ^ c)')' = ((b' v a') ^ (b ^ c)')'
8	3, 7	ax-r2 36	. . . . . . . . . 10 ((b ^ a) v (b ^ c)) = ((b' v a') ^ (b ^ c)')'
9	8	con2 67	. . . . . . . . 9 ((b ^ a) v (b ^ c))' = ((b' v a') ^ (b ^ c)')
10	9	lan 77	. . . . . . . 8 ((b ^ (a v c)) ^ ((b ^ a) v (b ^ c))') = ((b ^ (a v c)) ^ ((b' v a') ^ (b ^ c)'))
11		an4 86	. . . . . . . . 9 ((b ^ (a v c)) ^ ((b' v a') ^ (b ^ c)')) = ((b ^ (b' v a')) ^ ((a v c) ^ (b ^ c)'))
12		ax-a1 30	. . . . . . . . . . . . . 14 a = a''
13	12	ax-r1 35	. . . . . . . . . . . . 13 a'' = a
14		wwfh2.1	. . . . . . . . . . . . 13 a C b
15	13, 14	bctr 181	. . . . . . . . . . . 12 a'' C b
16	15	wwcom3ii 215	. . . . . . . . . . 11 (b ^ (b' v a')) = (b ^ a')
17		ancom 74	. . . . . . . . . . 11 (b ^ a') = (a' ^ b)
18	16, 17	ax-r2 36	. . . . . . . . . 10 (b ^ (b' v a')) = (a' ^ b)
19	18	ran 78	. . . . . . . . 9 ((b ^ (b' v a')) ^ ((a v c) ^ (b ^ c)')) = ((a' ^ b) ^ ((a v c) ^ (b ^ c)'))
20	11, 19	ax-r2 36	. . . . . . . 8 ((b ^ (a v c)) ^ ((b' v a') ^ (b ^ c)')) = ((a' ^ b) ^ ((a v c) ^ (b ^ c)'))
21	10, 20	ax-r2 36	. . . . . . 7 ((b ^ (a v c)) ^ ((b ^ a) v (b ^ c))') = ((a' ^ b) ^ ((a v c) ^ (b ^ c)'))
22		an4 86	. . . . . . 7 ((a' ^ b) ^ ((a v c) ^ (b ^ c)')) = ((a' ^ (a v c)) ^ (b ^ (b ^ c)'))
23	21, 22	ax-r2 36	. . . . . 6 ((b ^ (a v c)) ^ ((b ^ a) v (b ^ c))') = ((a' ^ (a v c)) ^ (b ^ (b ^ c)'))
24	12	ax-r5 38	. . . . . . . . 9 (a v c) = (a'' v c)
25	24	lan 77	. . . . . . . 8 (a' ^ (a v c)) = (a' ^ (a'' v c))
26		wwfh2.2	. . . . . . . . . 10 c' C a
27	26	comcom2 183	. . . . . . . . 9 c' C a'
28	27	wwcom3ii 215	. . . . . . . 8 (a' ^ (a'' v c)) = (a' ^ c)
29	25, 28	ax-r2 36	. . . . . . 7 (a' ^ (a v c)) = (a' ^ c)
30	29	ran 78	. . . . . 6 ((a' ^ (a v c)) ^ (b ^ (b ^ c)')) = ((a' ^ c) ^ (b ^ (b ^ c)'))
31	23, 30	ax-r2 36	. . . . 5 ((b ^ (a v c)) ^ ((b ^ a) v (b ^ c))') = ((a' ^ c) ^ (b ^ (b ^ c)'))
32		anass 76	. . . . 5 ((a' ^ c) ^ (b ^ (b ^ c)')) = (a' ^ (c ^ (b ^ (b ^ c)')))
33	31, 32	ax-r2 36	. . . 4 ((b ^ (a v c)) ^ ((b ^ a) v (b ^ c))') = (a' ^ (c ^ (b ^ (b ^ c)')))
34		anass 76	. . . . . . . 8 ((b ^ c) ^ (b ^ c)') = (b ^ (c ^ (b ^ c)'))
35	34	ax-r1 35	. . . . . . 7 (b ^ (c ^ (b ^ c)')) = ((b ^ c) ^ (b ^ c)')
36		an12 81	. . . . . . 7 (c ^ (b ^ (b ^ c)')) = (b ^ (c ^ (b ^ c)'))
37		dff 101	. . . . . . 7 0 = ((b ^ c) ^ (b ^ c)')
38	35, 36, 37	3tr1 63	. . . . . 6 (c ^ (b ^ (b ^ c)')) = 0
39	38	lan 77	. . . . 5 (a' ^ (c ^ (b ^ (b ^ c)'))) = (a' ^ 0)
40		an0 108	. . . . 5 (a' ^ 0) = 0
41	39, 40	ax-r2 36	. . . 4 (a' ^ (c ^ (b ^ (b ^ c)'))) = 0
42	33, 41	ax-r2 36	. . 3 ((b ^ (a v c)) ^ ((b ^ a) v (b ^ c))') = 0
43	2, 42	wwoml3 213	. 2 (((b ^ a) v (b ^ c)) == (b ^ (a v c))) = 1
44	1, 43	ax-r2 36	1 ((b ^ (a v c)) == ((b ^ a) v (b ^ c))) = 1

Syntax hints:   = wb 1   C wc 3  'wn 4   == tb 5   v wo 6   ^ wa 7  1wt 8  0wf 9

This theorem is referenced by:  wwfh4 219

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

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

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