Theorem u3lem6 749

Description: Lemma for unified implication study.

Assertion

Ref	Expression
u3lem6	(a ->3 (a ->3 (a ->3 b))) = (a ->3 (a ->3 b))

Proof of Theorem u3lem6

Step	Hyp	Ref	Expression
1		comi31 490	. . 3 a C (a ->3 (a ->3 b))
2	1	u3lemc4 685	. 2 (a ->3 (a ->3 (a ->3 b))) = (a_|_ v (a ->3 (a ->3 b)))
3		u3lem5 745	. . . 4 (a ->3 (a ->3 b)) = (a_|_ v b)
4	3	lor 66	. . 3 (a_|_ v (a ->3 (a ->3 b))) = (a_|_ v (a_|_ v b))
5		ax-a3 31	. . . . 5 ((a_|_ v a_|_) v b) = (a_|_ v (a_|_ v b))
6	5	ax-r1 34	. . . 4 (a_|_ v (a_|_ v b)) = ((a_|_ v a_|_) v b)
7		oridm 102	. . . . . 6 (a_|_ v a_|_) = a_|_
8	7	ax-r5 37	. . . . 5 ((a_|_ v a_|_) v b) = (a_|_ v b)
9	3	ax-r1 34	. . . . 5 (a_|_ v b) = (a ->3 (a ->3 b))
10	8, 9	ax-r2 35	. . . 4 ((a_|_ v a_|_) v b) = (a ->3 (a ->3 b))
11	6, 10	ax-r2 35	. . 3 (a_|_ v (a_|_ v b)) = (a ->3 (a ->3 b))
12	4, 11	ax-r2 35	. 2 (a_|_ v (a ->3 (a ->3 b))) = (a ->3 (a ->3 b))
13	2, 12	ax-r2 35	1 (a ->3 (a ->3 (a ->3 b))) = (a ->3 (a ->3 b))

Syntax hints:   = wb 1  _|_wn 4   v wo 6   ->3 wi3 15

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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