Theorem ud2lem1 563

Description: Lemma for unified disjunction. (Contributed by NM, 22-Nov-1997.)

Assertion

Ref	Expression
ud2lem1	((a →2 b) →2 (b →2 a)) = (a ∪ (a⊥ ∩ b⊥ ))

Proof of Theorem ud2lem1

Step	Hyp	Ref	Expression
1		df-i2 45	. 2 ((a →2 b) →2 (b →2 a)) = ((b →2 a) ∪ ((a →2 b)⊥ ∩ (b →2 a)⊥ ))
2		df-i2 45	. . . 4 (b →2 a) = (a ∪ (b⊥ ∩ a⊥ ))
3		ud2lem0c 278	. . . . 5 (a →2 b)⊥ = (b⊥ ∩ (a ∪ b))
4		ud2lem0c 278	. . . . 5 (b →2 a)⊥ = (a⊥ ∩ (b ∪ a))
5	3, 4	2an 79	. . . 4 ((a →2 b)⊥ ∩ (b →2 a)⊥ ) = ((b⊥ ∩ (a ∪ b)) ∩ (a⊥ ∩ (b ∪ a)))
6	2, 5	2or 72	. . 3 ((b →2 a) ∪ ((a →2 b)⊥ ∩ (b →2 a)⊥ )) = ((a ∪ (b⊥ ∩ a⊥ )) ∪ ((b⊥ ∩ (a ∪ b)) ∩ (a⊥ ∩ (b ∪ a))))
7		ancom 74	. . . . . 6 (b⊥ ∩ a⊥ ) = (a⊥ ∩ b⊥ )
8	7	lor 70	. . . . 5 (a ∪ (b⊥ ∩ a⊥ )) = (a ∪ (a⊥ ∩ b⊥ ))
9		dff 101	. . . . . . . 8 0 = ((b⊥ ∩ a⊥ ) ∩ (b⊥ ∩ a⊥ )⊥ )
10		oran 87	. . . . . . . . . 10 (b ∪ a) = (b⊥ ∩ a⊥ )⊥
11	10	ax-r1 35	. . . . . . . . 9 (b⊥ ∩ a⊥ )⊥ = (b ∪ a)
12	11	lan 77	. . . . . . . 8 ((b⊥ ∩ a⊥ ) ∩ (b⊥ ∩ a⊥ )⊥ ) = ((b⊥ ∩ a⊥ ) ∩ (b ∪ a))
13	9, 12	ax-r2 36	. . . . . . 7 0 = ((b⊥ ∩ a⊥ ) ∩ (b ∪ a))
14		anandir 115	. . . . . . . 8 ((b⊥ ∩ a⊥ ) ∩ (b ∪ a)) = ((b⊥ ∩ (b ∪ a)) ∩ (a⊥ ∩ (b ∪ a)))
15		ax-a2 31	. . . . . . . . . 10 (b ∪ a) = (a ∪ b)
16	15	lan 77	. . . . . . . . 9 (b⊥ ∩ (b ∪ a)) = (b⊥ ∩ (a ∪ b))
17	16	ran 78	. . . . . . . 8 ((b⊥ ∩ (b ∪ a)) ∩ (a⊥ ∩ (b ∪ a))) = ((b⊥ ∩ (a ∪ b)) ∩ (a⊥ ∩ (b ∪ a)))
18	14, 17	ax-r2 36	. . . . . . 7 ((b⊥ ∩ a⊥ ) ∩ (b ∪ a)) = ((b⊥ ∩ (a ∪ b)) ∩ (a⊥ ∩ (b ∪ a)))
19	13, 18	ax-r2 36	. . . . . 6 0 = ((b⊥ ∩ (a ∪ b)) ∩ (a⊥ ∩ (b ∪ a)))
20	19	ax-r1 35	. . . . 5 ((b⊥ ∩ (a ∪ b)) ∩ (a⊥ ∩ (b ∪ a))) = 0
21	8, 20	2or 72	. . . 4 ((a ∪ (b⊥ ∩ a⊥ )) ∪ ((b⊥ ∩ (a ∪ b)) ∩ (a⊥ ∩ (b ∪ a)))) = ((a ∪ (a⊥ ∩ b⊥ )) ∪ 0)
22		or0 102	. . . 4 ((a ∪ (a⊥ ∩ b⊥ )) ∪ 0) = (a ∪ (a⊥ ∩ b⊥ ))
23	21, 22	ax-r2 36	. . 3 ((a ∪ (b⊥ ∩ a⊥ )) ∪ ((b⊥ ∩ (a ∪ b)) ∩ (a⊥ ∩ (b ∪ a)))) = (a ∪ (a⊥ ∩ b⊥ ))
24	6, 23	ax-r2 36	. 2 ((b →2 a) ∪ ((a →2 b)⊥ ∩ (b →2 a)⊥ )) = (a ∪ (a⊥ ∩ b⊥ ))
25	1, 24	ax-r2 36	1 ((a →2 b) →2 (b →2 a)) = (a ∪ (a⊥ ∩ 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-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

This theorem is referenced by:  ud2  596