Theorem i3lem3 488

Description: Lemma for Kalmbach implication.

Hypothesis

Ref	Expression
i3lem.1	(a ->3 b) = 1

Assertion

Ref	Expression
i3lem3	((a_|_ v b) ^ b_|_) = (a_|_ ^ b_|_)

Proof of Theorem i3lem3

Step	Hyp	Ref	Expression
1		omlan 430	. 2 (b_|_ ^ (b v (b_|_ ^ a_|_))) = (b_|_ ^ a_|_)
2		ancom 68	. . 3 ((a_|_ v b) ^ b_|_) = (b_|_ ^ (a_|_ v b))
3		ax-a2 30	. . . . 5 (a_|_ v b) = (b v a_|_)
4		ax-a3 31	. . . . . . 7 ((b v (a_|_ ^ b)) v (a_|_ ^ b_|_)) = (b v ((a_|_ ^ b) v (a_|_ ^ b_|_)))
5	4	ax-r1 34	. . . . . 6 (b v ((a_|_ ^ b) v (a_|_ ^ b_|_))) = ((b v (a_|_ ^ b)) v (a_|_ ^ b_|_))
6		i3lem.1	. . . . . . . 8 (a ->3 b) = 1
7	6	i3lem1 486	. . . . . . 7 ((a_|_ ^ b) v (a_|_ ^ b_|_)) = a_|_
8	7	lor 66	. . . . . 6 (b v ((a_|_ ^ b) v (a_|_ ^ b_|_))) = (b v a_|_)
9		ancom 68	. . . . . . . . 9 (a_|_ ^ b) = (b ^ a_|_)
10	9	lor 66	. . . . . . . 8 (b v (a_|_ ^ b)) = (b v (b ^ a_|_))
11		a5b 112	. . . . . . . 8 (b v (b ^ a_|_)) = b
12	10, 11	ax-r2 35	. . . . . . 7 (b v (a_|_ ^ b)) = b
13		ancom 68	. . . . . . 7 (a_|_ ^ b_|_) = (b_|_ ^ a_|_)
14	12, 13	2or 67	. . . . . 6 ((b v (a_|_ ^ b)) v (a_|_ ^ b_|_)) = (b v (b_|_ ^ a_|_))
15	5, 8, 14	3tr2 61	. . . . 5 (b v a_|_) = (b v (b_|_ ^ a_|_))
16	3, 15	ax-r2 35	. . . 4 (a_|_ v b) = (b v (b_|_ ^ a_|_))
17	16	lan 70	. . 3 (b_|_ ^ (a_|_ v b)) = (b_|_ ^ (b v (b_|_ ^ a_|_)))
18	2, 17	ax-r2 35	. 2 ((a_|_ v b) ^ b_|_) = (b_|_ ^ (b v (b_|_ ^ a_|_)))
19	1, 18, 13	3tr1 60	1 ((a_|_ v b) ^ b_|_) = (a_|_ ^ b_|_)

Syntax hints:   = wb 1  _|_wn 4   v wo 6   ^ wa 7  1wt 9   ->3 wi3 15

This theorem is referenced by:  i3le 497

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-i3 45  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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