Theorem ud2lem2 564

Description: Lemma for unified disjunction. (Contributed by NM, 23-Nov-1997.)

Assertion

Ref	Expression
ud2lem2	((a ∪ (a⊥ ∩ b⊥ )) →2 a) = (a ∪ b)

Proof of Theorem ud2lem2

Step	Hyp	Ref	Expression
1		df-i2 45	. 2 ((a ∪ (a⊥ ∩ b⊥ )) →2 a) = (a ∪ ((a ∪ (a⊥ ∩ b⊥ ))⊥ ∩ a⊥ ))
2		oran 87	. . . . . . 7 ((a ∪ (a⊥ ∩ b⊥ )) ∪ a) = ((a ∪ (a⊥ ∩ b⊥ ))⊥ ∩ a⊥ )⊥
3	2	con2 67	. . . . . 6 ((a ∪ (a⊥ ∩ b⊥ )) ∪ a)⊥ = ((a ∪ (a⊥ ∩ b⊥ ))⊥ ∩ a⊥ )
4	3	ax-r1 35	. . . . 5 ((a ∪ (a⊥ ∩ b⊥ ))⊥ ∩ a⊥ ) = ((a ∪ (a⊥ ∩ b⊥ )) ∪ a)⊥
5		oran 87	. . . . . . . . . . . . 13 (a ∪ b) = (a⊥ ∩ b⊥ )⊥
6	5	con2 67	. . . . . . . . . . . 12 (a ∪ b)⊥ = (a⊥ ∩ b⊥ )
7	6	ax-r1 35	. . . . . . . . . . 11 (a⊥ ∩ b⊥ ) = (a ∪ b)⊥
8	7	lor 70	. . . . . . . . . 10 (a ∪ (a⊥ ∩ b⊥ )) = (a ∪ (a ∪ b)⊥ )
9		anor2 89	. . . . . . . . . . . 12 (a⊥ ∩ (a ∪ b)) = (a ∪ (a ∪ b)⊥ )⊥
10	9	ax-r1 35	. . . . . . . . . . 11 (a ∪ (a ∪ b)⊥ )⊥ = (a⊥ ∩ (a ∪ b))
11	10	con3 68	. . . . . . . . . 10 (a ∪ (a ∪ b)⊥ ) = (a⊥ ∩ (a ∪ b))⊥
12	8, 11	ax-r2 36	. . . . . . . . 9 (a ∪ (a⊥ ∩ b⊥ )) = (a⊥ ∩ (a ∪ b))⊥
13	12	con2 67	. . . . . . . 8 (a ∪ (a⊥ ∩ b⊥ ))⊥ = (a⊥ ∩ (a ∪ b))
14	13	ran 78	. . . . . . 7 ((a ∪ (a⊥ ∩ b⊥ ))⊥ ∩ a⊥ ) = ((a⊥ ∩ (a ∪ b)) ∩ a⊥ )
15		an32 83	. . . . . . . 8 ((a⊥ ∩ (a ∪ b)) ∩ a⊥ ) = ((a⊥ ∩ a⊥ ) ∩ (a ∪ b))
16		anidm 111	. . . . . . . . 9 (a⊥ ∩ a⊥ ) = a⊥
17	16	ran 78	. . . . . . . 8 ((a⊥ ∩ a⊥ ) ∩ (a ∪ b)) = (a⊥ ∩ (a ∪ b))
18	15, 17	ax-r2 36	. . . . . . 7 ((a⊥ ∩ (a ∪ b)) ∩ a⊥ ) = (a⊥ ∩ (a ∪ b))
19	14, 18	ax-r2 36	. . . . . 6 ((a ∪ (a⊥ ∩ b⊥ ))⊥ ∩ a⊥ ) = (a⊥ ∩ (a ∪ b))
20	3, 19	ax-r2 36	. . . . 5 ((a ∪ (a⊥ ∩ b⊥ )) ∪ a)⊥ = (a⊥ ∩ (a ∪ b))
21	4, 20	ax-r2 36	. . . 4 ((a ∪ (a⊥ ∩ b⊥ ))⊥ ∩ a⊥ ) = (a⊥ ∩ (a ∪ b))
22	21	lor 70	. . 3 (a ∪ ((a ∪ (a⊥ ∩ b⊥ ))⊥ ∩ a⊥ )) = (a ∪ (a⊥ ∩ (a ∪ b)))
23		oml 445	. . 3 (a ∪ (a⊥ ∩ (a ∪ b))) = (a ∪ b)
24	22, 23	ax-r2 36	. 2 (a ∪ ((a ∪ (a⊥ ∩ b⊥ ))⊥ ∩ a⊥ )) = (a ∪ b)
25	1, 24	ax-r2 36	1 ((a ∪ (a⊥ ∩ b⊥ )) →2 a) = (a ∪ b)

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →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  ax-r3 439

This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i2 45

This theorem is referenced by:  ud2  596