Theorem u2lem8 782

Description: Lemma for unified implication study. (Contributed by NM, 24-Dec-1997.)

Assertion

Ref	Expression
u2lem8	(a⊥ →2 (a →2 (a⊥ →2 b))) = (a →2 (a⊥ →2 b))

Proof of Theorem u2lem8

Step	Hyp	Ref	Expression
1		df-i2 45	. 2 (a⊥ →2 (a →2 (a⊥ →2 b))) = ((a →2 (a⊥ →2 b)) ∪ (a⊥ ⊥ ∩ (a →2 (a⊥ →2 b))⊥ ))
2		u2lem7 773	. . . 4 (a →2 (a⊥ →2 b)) = (((a ∩ b⊥ ) ∪ (a⊥ ∩ b⊥ )) ∪ b)
3		ax-a1 30	. . . . . . 7 a = a⊥ ⊥
4	3	ax-r1 35	. . . . . 6 a⊥ ⊥ = a
5		u2lem7n 775	. . . . . 6 (a →2 (a⊥ →2 b))⊥ = (((a ∪ b) ∩ (a⊥ ∪ b)) ∩ b⊥ )
6	4, 5	2an 79	. . . . 5 (a⊥ ⊥ ∩ (a →2 (a⊥ →2 b))⊥ ) = (a ∩ (((a ∪ b) ∩ (a⊥ ∪ b)) ∩ b⊥ ))
7		an12 81	. . . . . 6 (a ∩ (((a ∪ b) ∩ (a⊥ ∪ b)) ∩ b⊥ )) = (((a ∪ b) ∩ (a⊥ ∪ b)) ∩ (a ∩ b⊥ ))
8		anass 76	. . . . . . 7 (((a ∪ b) ∩ (a⊥ ∪ b)) ∩ (a ∩ b⊥ )) = ((a ∪ b) ∩ ((a⊥ ∪ b) ∩ (a ∩ b⊥ )))
9		anor1 88	. . . . . . . . . . 11 (a ∩ b⊥ ) = (a⊥ ∪ b)⊥
10	9	lan 77	. . . . . . . . . 10 ((a⊥ ∪ b) ∩ (a ∩ b⊥ )) = ((a⊥ ∪ b) ∩ (a⊥ ∪ b)⊥ )
11		dff 101	. . . . . . . . . . 11 0 = ((a⊥ ∪ b) ∩ (a⊥ ∪ b)⊥ )
12	11	ax-r1 35	. . . . . . . . . 10 ((a⊥ ∪ b) ∩ (a⊥ ∪ b)⊥ ) = 0
13	10, 12	ax-r2 36	. . . . . . . . 9 ((a⊥ ∪ b) ∩ (a ∩ b⊥ )) = 0
14	13	lan 77	. . . . . . . 8 ((a ∪ b) ∩ ((a⊥ ∪ b) ∩ (a ∩ b⊥ ))) = ((a ∪ b) ∩ 0)
15		an0 108	. . . . . . . 8 ((a ∪ b) ∩ 0) = 0
16	14, 15	ax-r2 36	. . . . . . 7 ((a ∪ b) ∩ ((a⊥ ∪ b) ∩ (a ∩ b⊥ ))) = 0
17	8, 16	ax-r2 36	. . . . . 6 (((a ∪ b) ∩ (a⊥ ∪ b)) ∩ (a ∩ b⊥ )) = 0
18	7, 17	ax-r2 36	. . . . 5 (a ∩ (((a ∪ b) ∩ (a⊥ ∪ b)) ∩ b⊥ )) = 0
19	6, 18	ax-r2 36	. . . 4 (a⊥ ⊥ ∩ (a →2 (a⊥ →2 b))⊥ ) = 0
20	2, 19	2or 72	. . 3 ((a →2 (a⊥ →2 b)) ∪ (a⊥ ⊥ ∩ (a →2 (a⊥ →2 b))⊥ )) = ((((a ∩ b⊥ ) ∪ (a⊥ ∩ b⊥ )) ∪ b) ∪ 0)
21		or0 102	. . . 4 ((((a ∩ b⊥ ) ∪ (a⊥ ∩ b⊥ )) ∪ b) ∪ 0) = (((a ∩ b⊥ ) ∪ (a⊥ ∩ b⊥ )) ∪ b)
22	2	ax-r1 35	. . . 4 (((a ∩ b⊥ ) ∪ (a⊥ ∩ b⊥ )) ∪ b) = (a →2 (a⊥ →2 b))
23	21, 22	ax-r2 36	. . 3 ((((a ∩ b⊥ ) ∪ (a⊥ ∩ b⊥ )) ∪ b) ∪ 0) = (a →2 (a⊥ →2 b))
24	20, 23	ax-r2 36	. 2 ((a →2 (a⊥ →2 b)) ∪ (a⊥ ⊥ ∩ (a →2 (a⊥ →2 b))⊥ )) = (a →2 (a⊥ →2 b))
25	1, 24	ax-r2 36	1 (a⊥ →2 (a →2 (a⊥ →2 b))) = (a →2 (a⊥ →2 b))

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  0wf 9   →₂ wi2 13

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38

This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-i2 45  df-le1 130  df-le2 131

This theorem is referenced by: (None)