Theorem wcom3ii 401

Description: Lemma 3(ii) of Kalmbach 83 p. 23.

Hypothesis

Ref	Expression
wcomcom.1	C (a, b) = 1

Assertion

Ref	Expression
wcom3ii	((a ^ (a_|_ v b)) == (a ^ b)) = 1

Proof of Theorem wcom3ii

Step	Hyp	Ref	Expression
1		wcomcom.1	. . . . . 6 C (a, b) = 1
2	1	wcomcom 396	. . . . 5 C (b, a) = 1
3	2	wcomd 400	. . . 4 (b == ((b v a) ^ (b v a_|_))) = 1
4	3	wlan 352	. . 3 ((a ^ b) == (a ^ ((b v a) ^ (b v a_|_)))) = 1
5		anass 69	. . . . . 6 ((a ^ (b v a)) ^ (b v a_|_)) = (a ^ ((b v a) ^ (b v a_|_)))
6	5	bi1 110	. . . . 5 (((a ^ (b v a)) ^ (b v a_|_)) == (a ^ ((b v a) ^ (b v a_|_)))) = 1
7	6	wr1 189	. . . 4 ((a ^ ((b v a) ^ (b v a_|_))) == ((a ^ (b v a)) ^ (b v a_|_))) = 1
8		ax-a2 30	. . . . . . . 8 (b v a) = (a v b)
9	8	bi1 110	. . . . . . 7 ((b v a) == (a v b)) = 1
10	9	wlan 352	. . . . . 6 ((a ^ (b v a)) == (a ^ (a v b))) = 1
11		a5c 113	. . . . . . 7 (a ^ (a v b)) = a
12	11	bi1 110	. . . . . 6 ((a ^ (a v b)) == a) = 1
13	10, 12	wr2 353	. . . . 5 ((a ^ (b v a)) == a) = 1
14		ax-a2 30	. . . . . 6 (b v a_|_) = (a_|_ v b)
15	14	bi1 110	. . . . 5 ((b v a_|_) == (a_|_ v b)) = 1
16	13, 15	w2an 355	. . . 4 (((a ^ (b v a)) ^ (b v a_|_)) == (a ^ (a_|_ v b))) = 1
17	7, 16	wr2 353	. . 3 ((a ^ ((b v a) ^ (b v a_|_))) == (a ^ (a_|_ v b))) = 1
18	4, 17	wr2 353	. 2 ((a ^ b) == (a ^ (a_|_ v b))) = 1
19	18	wr1 189	1 ((a ^ (a_|_ v b)) == (a ^ b)) = 1

Syntax hints:   = wb 1  _|_wn 4   == tb 5   v wo 6   ^ wa 7  1wt 9   C wcmtr 28

This theorem is referenced by:  wfh1 405  wfh2 406

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-wom 343

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-i1 43  df-i2 44  df-le 121  df-le1 122  df-le2 123  df-cmtr 126

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