Theorem i1abs 783

Description: An absorption law for ->1.

Assertion

Ref	Expression
i1abs	((a ->1 b)_|_ v (a ^ b)) = a

Proof of Theorem i1abs

Step	Hyp	Ref	Expression
1		ud1lem0c 269	. . 3 (a ->1 b)_|_ = (a ^ (a_|_ v b_|_))
2	1	ax-r5 37	. 2 ((a ->1 b)_|_ v (a ^ b)) = ((a ^ (a_|_ v b_|_)) v (a ^ b))
3		comanr1 446	. . 3 a C (a ^ b)
4		comorr 176	. . . 4 a_|_ C (a_|_ v b_|_)
5	4	comcom6 441	. . 3 a C (a_|_ v b_|_)
6	3, 5	fh4r 458	. 2 ((a ^ (a_|_ v b_|_)) v (a ^ b)) = ((a v (a ^ b)) ^ ((a_|_ v b_|_) v (a ^ b)))
7		a5b 112	. . . 4 (a v (a ^ b)) = a
8		df-a 39	. . . . . 6 (a ^ b) = (a_|_ v b_|_)_|_
9	8	lor 66	. . . . 5 ((a_|_ v b_|_) v (a ^ b)) = ((a_|_ v b_|_) v (a_|_ v b_|_)_|_)
10		df-t 40	. . . . . 6 1 = ((a_|_ v b_|_) v (a_|_ v b_|_)_|_)
11	10	ax-r1 34	. . . . 5 ((a_|_ v b_|_) v (a_|_ v b_|_)_|_) = 1
12	9, 11	ax-r2 35	. . . 4 ((a_|_ v b_|_) v (a ^ b)) = 1
13	7, 12	2an 72	. . 3 ((a v (a ^ b)) ^ ((a_|_ v b_|_) v (a ^ b))) = (a ^ 1)
14		an1 98	. . 3 (a ^ 1) = a
15	13, 14	ax-r2 35	. 2 ((a v (a ^ b)) ^ ((a_|_ v b_|_) v (a ^ b))) = a
16	2, 6, 15	3tr 62	1 ((a ->1 b)_|_ v (a ^ b)) = a

Syntax hints:   = wb 1  _|_wn 4   v wo 6   ^ wa 7  1wt 9   ->1 wi1 13

This theorem is referenced by:  cancellem 873

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-r3 421

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-i1 43  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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