Theorem wcomdr 403

Description: Commutation dual. Kalmbach 83 p. 23.

Hypothesis

Ref	Expression
wcomdr.1	(a == ((a v b) ^ (a v b_|_))) = 1

Assertion

Ref	Expression
wcomdr	C (a, b) = 1

Proof of Theorem wcomdr

Step	Hyp	Ref	Expression
1		wcomdr.1	. . . . 5 (a == ((a v b) ^ (a v b_|_))) = 1
2		df-a 39	. . . . . . 7 ((a v b) ^ (a v b_|_)) = ((a v b)_|_ v (a v b_|_)_|_)_|_
3	2	bi1 110	. . . . . 6 (((a v b) ^ (a v b_|_)) == ((a v b)_|_ v (a v b_|_)_|_)_|_) = 1
4		oran 79	. . . . . . . . . 10 (a v b) = (a_|_ ^ b_|_)_|_
5	4	bi1 110	. . . . . . . . 9 ((a v b) == (a_|_ ^ b_|_)_|_) = 1
6	5	wcon2 200	. . . . . . . 8 ((a v b)_|_ == (a_|_ ^ b_|_)) = 1
7		oran 79	. . . . . . . . . 10 (a v b_|_) = (a_|_ ^ b_|__|_)_|_
8	7	bi1 110	. . . . . . . . 9 ((a v b_|_) == (a_|_ ^ b_|__|_)_|_) = 1
9	8	wcon2 200	. . . . . . . 8 ((a v b_|_)_|_ == (a_|_ ^ b_|__|_)) = 1
10	6, 9	w2or 354	. . . . . . 7 (((a v b)_|_ v (a v b_|_)_|_) == ((a_|_ ^ b_|_) v (a_|_ ^ b_|__|_))) = 1
11	10	wr4 191	. . . . . 6 (((a v b)_|_ v (a v b_|_)_|_)_|_ == ((a_|_ ^ b_|_) v (a_|_ ^ b_|__|_))_|_) = 1
12	3, 11	wr2 353	. . . . 5 (((a v b) ^ (a v b_|_)) == ((a_|_ ^ b_|_) v (a_|_ ^ b_|__|_))_|_) = 1
13	1, 12	wr2 353	. . . 4 (a == ((a_|_ ^ b_|_) v (a_|_ ^ b_|__|_))_|_) = 1
14	13	wcon2 200	. . 3 (a_|_ == ((a_|_ ^ b_|_) v (a_|_ ^ b_|__|_))) = 1
15	14	wdf-c1 365	. 2 C (a_|_, b_|_) = 1
16	15	wcomcom5 402	1 C (a, b) = 1

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

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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