Theorem ud1lem2 561

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

Assertion

Ref	Expression
ud1lem2	((a ∪ (a⊥ ∩ b⊥ )) →1 a) = (a ∪ b)

Proof of Theorem ud1lem2

Step	Hyp	Ref	Expression
1		df-i1 44	. 2 ((a ∪ (a⊥ ∩ b⊥ )) →1 a) = ((a ∪ (a⊥ ∩ b⊥ ))⊥ ∪ ((a ∪ (a⊥ ∩ b⊥ )) ∩ a))
2		comid 187	. . . 4 (a ∪ (a⊥ ∩ b⊥ )) C (a ∪ (a⊥ ∩ b⊥ ))
3	2	comcom3 454	. . 3 (a ∪ (a⊥ ∩ b⊥ ))⊥ C (a ∪ (a⊥ ∩ b⊥ ))
4		comor1 461	. . . 4 (a ∪ (a⊥ ∩ b⊥ )) C a
5	4	comcom3 454	. . 3 (a ∪ (a⊥ ∩ b⊥ ))⊥ C a
6	3, 5	fh3 471	. 2 ((a ∪ (a⊥ ∩ b⊥ ))⊥ ∪ ((a ∪ (a⊥ ∩ b⊥ )) ∩ a)) = (((a ∪ (a⊥ ∩ b⊥ ))⊥ ∪ (a ∪ (a⊥ ∩ b⊥ ))) ∩ ((a ∪ (a⊥ ∩ b⊥ ))⊥ ∪ a))
7		ancom 74	. . 3 (((a ∪ (a⊥ ∩ b⊥ ))⊥ ∪ (a ∪ (a⊥ ∩ b⊥ ))) ∩ ((a ∪ (a⊥ ∩ b⊥ ))⊥ ∪ a)) = (((a ∪ (a⊥ ∩ b⊥ ))⊥ ∪ a) ∩ ((a ∪ (a⊥ ∩ b⊥ ))⊥ ∪ (a ∪ (a⊥ ∩ b⊥ ))))
8		ax-a2 31	. . . . 5 ((a ∪ (a⊥ ∩ b⊥ ))⊥ ∪ (a ∪ (a⊥ ∩ b⊥ ))) = ((a ∪ (a⊥ ∩ b⊥ )) ∪ (a ∪ (a⊥ ∩ b⊥ ))⊥ )
9		df-t 41	. . . . . 6 1 = ((a ∪ (a⊥ ∩ b⊥ )) ∪ (a ∪ (a⊥ ∩ b⊥ ))⊥ )
10	9	ax-r1 35	. . . . 5 ((a ∪ (a⊥ ∩ b⊥ )) ∪ (a ∪ (a⊥ ∩ b⊥ ))⊥ ) = 1
11	8, 10	ax-r2 36	. . . 4 ((a ∪ (a⊥ ∩ b⊥ ))⊥ ∪ (a ∪ (a⊥ ∩ b⊥ ))) = 1
12	11	lan 77	. . 3 (((a ∪ (a⊥ ∩ b⊥ ))⊥ ∪ a) ∩ ((a ∪ (a⊥ ∩ b⊥ ))⊥ ∪ (a ∪ (a⊥ ∩ b⊥ )))) = (((a ∪ (a⊥ ∩ b⊥ ))⊥ ∪ a) ∩ 1)
13		an1 106	. . . 4 (((a ∪ (a⊥ ∩ b⊥ ))⊥ ∪ a) ∩ 1) = ((a ∪ (a⊥ ∩ b⊥ ))⊥ ∪ a)
14		oran 87	. . . . . . 7 (a ∪ (a⊥ ∩ b⊥ )) = (a⊥ ∩ (a⊥ ∩ b⊥ )⊥ )⊥
15		oran 87	. . . . . . . . . 10 (a ∪ b) = (a⊥ ∩ b⊥ )⊥
16	15	ax-r1 35	. . . . . . . . 9 (a⊥ ∩ b⊥ )⊥ = (a ∪ b)
17	16	lan 77	. . . . . . . 8 (a⊥ ∩ (a⊥ ∩ b⊥ )⊥ ) = (a⊥ ∩ (a ∪ b))
18	17	ax-r4 37	. . . . . . 7 (a⊥ ∩ (a⊥ ∩ b⊥ )⊥ )⊥ = (a⊥ ∩ (a ∪ b))⊥
19	14, 18	ax-r2 36	. . . . . 6 (a ∪ (a⊥ ∩ b⊥ )) = (a⊥ ∩ (a ∪ b))⊥
20	19	con2 67	. . . . 5 (a ∪ (a⊥ ∩ b⊥ ))⊥ = (a⊥ ∩ (a ∪ b))
21	20	ax-r5 38	. . . 4 ((a ∪ (a⊥ ∩ b⊥ ))⊥ ∪ a) = ((a⊥ ∩ (a ∪ b)) ∪ a)
22		ax-a2 31	. . . . 5 ((a⊥ ∩ (a ∪ b)) ∪ a) = (a ∪ (a⊥ ∩ (a ∪ b)))
23		oml 445	. . . . 5 (a ∪ (a⊥ ∩ (a ∪ b))) = (a ∪ b)
24	22, 23	ax-r2 36	. . . 4 ((a⊥ ∩ (a ∪ b)) ∪ a) = (a ∪ b)
25	13, 21, 24	3tr 65	. . 3 (((a ∪ (a⊥ ∩ b⊥ ))⊥ ∪ a) ∩ 1) = (a ∪ b)
26	7, 12, 25	3tr 65	. 2 (((a ∪ (a⊥ ∩ b⊥ ))⊥ ∪ (a ∪ (a⊥ ∩ b⊥ ))) ∩ ((a ∪ (a⊥ ∩ b⊥ ))⊥ ∪ a)) = (a ∪ b)
27	1, 6, 26	3tr 65	1 ((a ∪ (a⊥ ∩ b⊥ )) →1 a) = (a ∪ b)

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   →₁ wi1 12

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-i1 44  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by:  ud1  595