Theorem wwcom3ii 207

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

Hypothesis

Ref	Expression
wwcom3ii.1	b_|_ C a

Assertion

Ref	Expression
wwcom3ii	(a ^ (a_|_ v b)) = (a ^ b)

Proof of Theorem wwcom3ii

Step	Hyp	Ref	Expression
1		wwcom3ii.1	. . . . 5 b_|_ C a
2	1	wwcomd 206	. . . 4 b = ((b v a) ^ (b v a_|_))
3	2	lan 70	. . 3 (a ^ b) = (a ^ ((b v a) ^ (b v a_|_)))
4		anass 69	. . . . 5 ((a ^ (b v a)) ^ (b v a_|_)) = (a ^ ((b v a) ^ (b v a_|_)))
5	4	ax-r1 34	. . . 4 (a ^ ((b v a) ^ (b v a_|_))) = ((a ^ (b v a)) ^ (b v a_|_))
6		ax-a2 30	. . . . . . 7 (b v a) = (a v b)
7	6	lan 70	. . . . . 6 (a ^ (b v a)) = (a ^ (a v b))
8		a5c 113	. . . . . 6 (a ^ (a v b)) = a
9	7, 8	ax-r2 35	. . . . 5 (a ^ (b v a)) = a
10		ax-a2 30	. . . . 5 (b v a_|_) = (a_|_ v b)
11	9, 10	2an 72	. . . 4 ((a ^ (b v a)) ^ (b v a_|_)) = (a ^ (a_|_ v b))
12	5, 11	ax-r2 35	. . 3 (a ^ ((b v a) ^ (b v a_|_))) = (a ^ (a_|_ v b))
13	3, 12	ax-r2 35	. 2 (a ^ b) = (a ^ (a_|_ v b))
14	13	ax-r1 34	1 (a ^ (a_|_ v b)) = (a ^ b)

Syntax hints:   = wb 1   C wc 3  _|_wn 4   v wo 6   ^ wa 7

This theorem is referenced by:  wwfh1 208  wwfh2 209

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37

This theorem depends on definitions:  df-a 39  df-c2 125

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