Theorem wwcomd 206

Description: Commutation dual (weak). Kalmbach 83 p. 23.

Hypothesis

Ref	Expression
wwcomd.1	a_|_ C b

Assertion

Ref	Expression
wwcomd	a = ((a v b) ^ (a v b_|_))

Proof of Theorem wwcomd

Step	Hyp	Ref	Expression
1		wwcomd.1	. . . 4 a_|_ C b
2	1	df-c2 125	. . 3 a_|_ = ((a_|_ ^ b) v (a_|_ ^ b_|_))
3		oran 79	. . . 4 ((a_|_ ^ b_|_) v (a_|_ ^ b)) = ((a_|_ ^ b_|_)_|_ ^ (a_|_ ^ b)_|_)_|_
4		ax-a2 30	. . . 4 ((a_|_ ^ b) v (a_|_ ^ b_|_)) = ((a_|_ ^ b_|_) v (a_|_ ^ b))
5		oran 79	. . . . . 6 (a v b) = (a_|_ ^ b_|_)_|_
6		anor2 81	. . . . . . . 8 (a_|_ ^ b) = (a v b_|_)_|_
7	6	ax-r1 34	. . . . . . 7 (a v b_|_)_|_ = (a_|_ ^ b)
8	7	con3 65	. . . . . 6 (a v b_|_) = (a_|_ ^ b)_|_
9	5, 8	2an 72	. . . . 5 ((a v b) ^ (a v b_|_)) = ((a_|_ ^ b_|_)_|_ ^ (a_|_ ^ b)_|_)
10	9	ax-r4 36	. . . 4 ((a v b) ^ (a v b_|_))_|_ = ((a_|_ ^ b_|_)_|_ ^ (a_|_ ^ b)_|_)_|_
11	3, 4, 10	3tr1 60	. . 3 ((a_|_ ^ b) v (a_|_ ^ b_|_)) = ((a v b) ^ (a v b_|_))_|_
12	2, 11	ax-r2 35	. 2 a_|_ = ((a v b) ^ (a v b_|_))_|_
13	12	con1 63	1 a = ((a v b) ^ (a v b_|_))

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

This theorem is referenced by:  wwcom3ii 207

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

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

metamath.org