Theorem ud2lem3 565

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

Assertion

Ref	Expression
ud2lem3	((a →2 b) →2 (a ∪ b)) = (a ∪ b)

Proof of Theorem ud2lem3

Step	Hyp	Ref	Expression
1		df-i2 45	. 2 ((a →2 b) →2 (a ∪ b)) = ((a ∪ b) ∪ ((a →2 b)⊥ ∩ (a ∪ b)⊥ ))
2		ud2lem0c 278	. . . . 5 (a →2 b)⊥ = (b⊥ ∩ (a ∪ b))
3	2	ran 78	. . . 4 ((a →2 b)⊥ ∩ (a ∪ b)⊥ ) = ((b⊥ ∩ (a ∪ b)) ∩ (a ∪ b)⊥ )
4	3	lor 70	. . 3 ((a ∪ b) ∪ ((a →2 b)⊥ ∩ (a ∪ b)⊥ )) = ((a ∪ b) ∪ ((b⊥ ∩ (a ∪ b)) ∩ (a ∪ b)⊥ ))
5		coman2 186	. . . . . 6 (b⊥ ∩ (a ∪ b)) C (a ∪ b)
6	5	comcom 453	. . . . 5 (a ∪ b) C (b⊥ ∩ (a ∪ b))
7		comid 187	. . . . . 6 (a ∪ b) C (a ∪ b)
8	7	comcom2 183	. . . . 5 (a ∪ b) C (a ∪ b)⊥
9	6, 8	fh3 471	. . . 4 ((a ∪ b) ∪ ((b⊥ ∩ (a ∪ b)) ∩ (a ∪ b)⊥ )) = (((a ∪ b) ∪ (b⊥ ∩ (a ∪ b))) ∩ ((a ∪ b) ∪ (a ∪ b)⊥ ))
10		ancom 74	. . . . . . 7 (b⊥ ∩ (a ∪ b)) = ((a ∪ b) ∩ b⊥ )
11	10	lor 70	. . . . . 6 ((a ∪ b) ∪ (b⊥ ∩ (a ∪ b))) = ((a ∪ b) ∪ ((a ∪ b) ∩ b⊥ ))
12		df-t 41	. . . . . . 7 1 = ((a ∪ b) ∪ (a ∪ b)⊥ )
13	12	ax-r1 35	. . . . . 6 ((a ∪ b) ∪ (a ∪ b)⊥ ) = 1
14	11, 13	2an 79	. . . . 5 (((a ∪ b) ∪ (b⊥ ∩ (a ∪ b))) ∩ ((a ∪ b) ∪ (a ∪ b)⊥ )) = (((a ∪ b) ∪ ((a ∪ b) ∩ b⊥ )) ∩ 1)
15		an1 106	. . . . . 6 (((a ∪ b) ∪ ((a ∪ b) ∩ b⊥ )) ∩ 1) = ((a ∪ b) ∪ ((a ∪ b) ∩ b⊥ ))
16		orabs 120	. . . . . 6 ((a ∪ b) ∪ ((a ∪ b) ∩ b⊥ )) = (a ∪ b)
17	15, 16	ax-r2 36	. . . . 5 (((a ∪ b) ∪ ((a ∪ b) ∩ b⊥ )) ∩ 1) = (a ∪ b)
18	14, 17	ax-r2 36	. . . 4 (((a ∪ b) ∪ (b⊥ ∩ (a ∪ b))) ∩ ((a ∪ b) ∪ (a ∪ b)⊥ )) = (a ∪ b)
19	9, 18	ax-r2 36	. . 3 ((a ∪ b) ∪ ((b⊥ ∩ (a ∪ b)) ∩ (a ∪ b)⊥ )) = (a ∪ b)
20	4, 19	ax-r2 36	. 2 ((a ∪ b) ∪ ((a →2 b)⊥ ∩ (a ∪ b)⊥ )) = (a ∪ b)
21	1, 20	ax-r2 36	1 ((a →2 b) →2 (a ∪ b)) = (a ∪ b)

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   →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  ax-r3 439

This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i2 45  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by:  ud2  596