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⊥ ∪ (a →3 (a →3 b)))
3		u3lem5 745	. . . 4 (a →3 (a →3 b)) = (a⊥ ∪ b)
4	3	lor 66	. . 3 (a⊥ ∪ (a →3 (a →3 b))) = (a⊥ ∪ (a⊥ ∪ b))
5		ax-a3 31	. . . . 5 ((a⊥ ∪ a⊥ ) ∪ b) = (a⊥ ∪ (a⊥ ∪ b))
6	5	ax-r1 34	. . . 4 (a⊥ ∪ (a⊥ ∪ b)) = ((a⊥ ∪ a⊥ ) ∪ b)
7		oridm 102	. . . . . 6 (a⊥ ∪ a⊥ ) = a⊥
8	7	ax-r5 37	. . . . 5 ((a⊥ ∪ a⊥ ) ∪ b) = (a⊥ ∪ b)
9	3	ax-r1 34	. . . . 5 (a⊥ ∪ b) = (a →3 (a →3 b))
10	8, 9	ax-r2 35	. . . 4 ((a⊥ ∪ a⊥ ) ∪ b) = (a →3 (a →3 b))
11	6, 10	ax-r2 35	. . 3 (a⊥ ∪ (a⊥ ∪ b)) = (a →3 (a →3 b))
12	4, 11	ax-r2 35	. 2 (a⊥ ∪ (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   ∪ 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

metamath.org