Theorem comdr 448

Description: Commutation dual. Kalmbach 83 p. 23.

Hypothesis

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

Assertion

Ref	Expression
comdr	a C b

Proof of Theorem comdr

Step	Hyp	Ref	Expression
1		comdr.1	. . . . 5 a = ((a v b) ^ (a v b_|_))
2		df-a 39	. . . . . 6 ((a v b) ^ (a v b_|_)) = ((a v b)_|_ v (a v b_|_)_|_)_|_
3		oran 79	. . . . . . . . 9 (a v b) = (a_|_ ^ b_|_)_|_
4	3	con2 64	. . . . . . . 8 (a v b)_|_ = (a_|_ ^ b_|_)
5		oran 79	. . . . . . . . 9 (a v b_|_) = (a_|_ ^ b_|__|_)_|_
6	5	con2 64	. . . . . . . 8 (a v b_|_)_|_ = (a_|_ ^ b_|__|_)
7	4, 6	2or 67	. . . . . . 7 ((a v b)_|_ v (a v b_|_)_|_) = ((a_|_ ^ b_|_) v (a_|_ ^ b_|__|_))
8	7	ax-r4 36	. . . . . 6 ((a v b)_|_ v (a v b_|_)_|_)_|_ = ((a_|_ ^ b_|_) v (a_|_ ^ b_|__|_))_|_
9	2, 8	ax-r2 35	. . . . 5 ((a v b) ^ (a v b_|_)) = ((a_|_ ^ b_|_) v (a_|_ ^ b_|__|_))_|_
10	1, 9	ax-r2 35	. . . 4 a = ((a_|_ ^ b_|_) v (a_|_ ^ b_|__|_))_|_
11	10	con2 64	. . 3 a_|_ = ((a_|_ ^ b_|_) v (a_|_ ^ b_|__|_))
12	11	df-c1 124	. 2 a_|_ C b_|_
13	12	comcom5 440	1 a C b

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

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  ax-r3 421

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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