Theorem wwfh1 208

Description: Foulis-Holland Theorem (weak).

Hypotheses

Ref	Expression
wwfh.1	b C a
wwfh.2	c C a

Assertion

Ref	Expression
wwfh1	((a ^ (b v c)) == ((a ^ b) v (a ^ c))) = 1

Proof of Theorem wwfh1

Step	Hyp	Ref	Expression
1		bicom 88	. 2 ((a ^ (b v c)) == ((a ^ b) v (a ^ c))) = (((a ^ b) v (a ^ c)) == (a ^ (b v c)))
2		ledi 166	. . 3 ((a ^ b) v (a ^ c)) =< (a ^ (b v c))
3		ancom 68	. . . . . 6 (a ^ (b v c)) = ((b v c) ^ a)
4		df-a 39	. . . . . . . . 9 (a ^ b) = (a_|_ v b_|_)_|_
5		df-a 39	. . . . . . . . 9 (a ^ c) = (a_|_ v c_|_)_|_
6	4, 5	2or 67	. . . . . . . 8 ((a ^ b) v (a ^ c)) = ((a_|_ v b_|_)_|_ v (a_|_ v c_|_)_|_)
7		df-a 39	. . . . . . . . . 10 ((a_|_ v b_|_) ^ (a_|_ v c_|_)) = ((a_|_ v b_|_)_|_ v (a_|_ v c_|_)_|_)_|_
8	7	ax-r1 34	. . . . . . . . 9 ((a_|_ v b_|_)_|_ v (a_|_ v c_|_)_|_)_|_ = ((a_|_ v b_|_) ^ (a_|_ v c_|_))
9	8	con3 65	. . . . . . . 8 ((a_|_ v b_|_)_|_ v (a_|_ v c_|_)_|_) = ((a_|_ v b_|_) ^ (a_|_ v c_|_))_|_
10	6, 9	ax-r2 35	. . . . . . 7 ((a ^ b) v (a ^ c)) = ((a_|_ v b_|_) ^ (a_|_ v c_|_))_|_
11	10	con2 64	. . . . . 6 ((a ^ b) v (a ^ c))_|_ = ((a_|_ v b_|_) ^ (a_|_ v c_|_))
12	3, 11	2an 72	. . . . 5 ((a ^ (b v c)) ^ ((a ^ b) v (a ^ c))_|_) = (((b v c) ^ a) ^ ((a_|_ v b_|_) ^ (a_|_ v c_|_)))
13		anass 69	. . . . . . 7 (((b v c) ^ a) ^ ((a_|_ v b_|_) ^ (a_|_ v c_|_))) = ((b v c) ^ (a ^ ((a_|_ v b_|_) ^ (a_|_ v c_|_))))
14		ax-a1 29	. . . . . . . . . . . . 13 b = b_|__|_
15	14	ax-r1 34	. . . . . . . . . . . 12 b_|__|_ = b
16		wwfh.1	. . . . . . . . . . . 12 b C a
17	15, 16	bctr 173	. . . . . . . . . . 11 b_|__|_ C a
18	17	wwcom3ii 207	. . . . . . . . . 10 (a ^ (a_|_ v b_|_)) = (a ^ b_|_)
19		ax-a1 29	. . . . . . . . . . . . 13 c = c_|__|_
20	19	ax-r1 34	. . . . . . . . . . . 12 c_|__|_ = c
21		wwfh.2	. . . . . . . . . . . 12 c C a
22	20, 21	bctr 173	. . . . . . . . . . 11 c_|__|_ C a
23	22	wwcom3ii 207	. . . . . . . . . 10 (a ^ (a_|_ v c_|_)) = (a ^ c_|_)
24	18, 23	2an 72	. . . . . . . . 9 ((a ^ (a_|_ v b_|_)) ^ (a ^ (a_|_ v c_|_))) = ((a ^ b_|_) ^ (a ^ c_|_))
25		anandi 106	. . . . . . . . 9 (a ^ ((a_|_ v b_|_) ^ (a_|_ v c_|_))) = ((a ^ (a_|_ v b_|_)) ^ (a ^ (a_|_ v c_|_)))
26		anandi 106	. . . . . . . . 9 (a ^ (b_|_ ^ c_|_)) = ((a ^ b_|_) ^ (a ^ c_|_))
27	24, 25, 26	3tr1 60	. . . . . . . 8 (a ^ ((a_|_ v b_|_) ^ (a_|_ v c_|_))) = (a ^ (b_|_ ^ c_|_))
28	27	lan 70	. . . . . . 7 ((b v c) ^ (a ^ ((a_|_ v b_|_) ^ (a_|_ v c_|_)))) = ((b v c) ^ (a ^ (b_|_ ^ c_|_)))
29	13, 28	ax-r2 35	. . . . . 6 (((b v c) ^ a) ^ ((a_|_ v b_|_) ^ (a_|_ v c_|_))) = ((b v c) ^ (a ^ (b_|_ ^ c_|_)))
30		an12 74	. . . . . 6 ((b v c) ^ (a ^ (b_|_ ^ c_|_))) = (a ^ ((b v c) ^ (b_|_ ^ c_|_)))
31	29, 30	ax-r2 35	. . . . 5 (((b v c) ^ a) ^ ((a_|_ v b_|_) ^ (a_|_ v c_|_))) = (a ^ ((b v c) ^ (b_|_ ^ c_|_)))
32	12, 31	ax-r2 35	. . . 4 ((a ^ (b v c)) ^ ((a ^ b) v (a ^ c))_|_) = (a ^ ((b v c) ^ (b_|_ ^ c_|_)))
33		oran 79	. . . . . . . . . 10 (b v c) = (b_|_ ^ c_|_)_|_
34	33	ax-r1 34	. . . . . . . . 9 (b_|_ ^ c_|_)_|_ = (b v c)
35	34	con3 65	. . . . . . . 8 (b_|_ ^ c_|_) = (b v c)_|_
36	35	lan 70	. . . . . . 7 ((b v c) ^ (b_|_ ^ c_|_)) = ((b v c) ^ (b v c)_|_)
37		dff 93	. . . . . . . 8 0 = ((b v c) ^ (b v c)_|_)
38	37	ax-r1 34	. . . . . . 7 ((b v c) ^ (b v c)_|_) = 0
39	36, 38	ax-r2 35	. . . . . 6 ((b v c) ^ (b_|_ ^ c_|_)) = 0
40	39	lan 70	. . . . 5 (a ^ ((b v c) ^ (b_|_ ^ c_|_))) = (a ^ 0)
41		an0 100	. . . . 5 (a ^ 0) = 0
42	40, 41	ax-r2 35	. . . 4 (a ^ ((b v c) ^ (b_|_ ^ c_|_))) = 0
43	32, 42	ax-r2 35	. . 3 ((a ^ (b v c)) ^ ((a ^ b) v (a ^ c))_|_) = 0
44	2, 43	wwoml3 205	. 2 (((a ^ b) v (a ^ c)) == (a ^ (b v c))) = 1
45	1, 44	ax-r2 35	1 ((a ^ (b v c)) == ((a ^ b) v (a ^ c))) = 1

Syntax hints:   = wb 1   C wc 3  _|_wn 4   == tb 5   v wo 6   ^ wa 7  1wt 9  0wf 10

This theorem is referenced by:  wwfh3 210

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

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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