Theorem ud1lem2 543

Description: Lemma for unified disjunction.

Assertion

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

Proof of Theorem ud1lem2

Step	Hyp	Ref	Expression
1		df-i1 43	. 2 ((a ∪ (a⊥ ∩ b⊥ )) →1 a) = ((a ∪ (a⊥ ∩ b⊥ ))⊥ ∪ ((a ∪ (a⊥ ∩ b⊥ )) ∩ a))
2		comid 179	. . . . 5 (a ∪ (a⊥ ∩ b⊥ )) C (a ∪ (a⊥ ∩ b⊥ ))
3	2	comcom3 436	. . . 4 (a ∪ (a⊥ ∩ b⊥ ))⊥ C (a ∪ (a⊥ ∩ b⊥ ))
4		comor1 443	. . . . 5 (a ∪ (a⊥ ∩ b⊥ )) C a
5	4	comcom3 436	. . . 4 (a ∪ (a⊥ ∩ b⊥ ))⊥ C a
6	3, 5	fh3 453	. . 3 ((a ∪ (a⊥ ∩ b⊥ ))⊥ ∪ ((a ∪ (a⊥ ∩ b⊥ )) ∩ a)) = (((a ∪ (a⊥ ∩ b⊥ ))⊥ ∪ (a ∪ (a⊥ ∩ b⊥ ))) ∩ ((a ∪ (a⊥ ∩ b⊥ ))⊥ ∪ a))
7		ancom 68	. . . 4 (((a ∪ (a⊥ ∩ b⊥ ))⊥ ∪ (a ∪ (a⊥ ∩ b⊥ ))) ∩ ((a ∪ (a⊥ ∩ b⊥ ))⊥ ∪ a)) = (((a ∪ (a⊥ ∩ b⊥ ))⊥ ∪ a) ∩ ((a ∪ (a⊥ ∩ b⊥ ))⊥ ∪ (a ∪ (a⊥ ∩ b⊥ ))))
8		ax-a2 30	. . . . . . 7 ((a ∪ (a⊥ ∩ b⊥ ))⊥ ∪ (a ∪ (a⊥ ∩ b⊥ ))) = ((a ∪ (a⊥ ∩ b⊥ )) ∪ (a ∪ (a⊥ ∩ b⊥ ))⊥ )
9		df-t 40	. . . . . . . 8 1 = ((a ∪ (a⊥ ∩ b⊥ )) ∪ (a ∪ (a⊥ ∩ b⊥ ))⊥ )
10	9	ax-r1 34	. . . . . . 7 ((a ∪ (a⊥ ∩ b⊥ )) ∪ (a ∪ (a⊥ ∩ b⊥ ))⊥ ) = 1
11	8, 10	ax-r2 35	. . . . . 6 ((a ∪ (a⊥ ∩ b⊥ ))⊥ ∪ (a ∪ (a⊥ ∩ b⊥ ))) = 1
12	11	lan 70	. . . . 5 (((a ∪ (a⊥ ∩ b⊥ ))⊥ ∪ a) ∩ ((a ∪ (a⊥ ∩ b⊥ ))⊥ ∪ (a ∪ (a⊥ ∩ b⊥ )))) = (((a ∪ (a⊥ ∩ b⊥ ))⊥ ∪ a) ∩ 1)
13		an1 98	. . . . . 6 (((a ∪ (a⊥ ∩ b⊥ ))⊥ ∪ a) ∩ 1) = ((a ∪ (a⊥ ∩ b⊥ ))⊥ ∪ a)
14		oran 79	. . . . . . . . . 10 (a ∪ (a⊥ ∩ b⊥ )) = (a⊥ ∩ (a⊥ ∩ b⊥ )⊥ )⊥
15		oran 79	. . . . . . . . . . . . 13 (a ∪ b) = (a⊥ ∩ b⊥ )⊥
16	15	ax-r1 34	. . . . . . . . . . . 12 (a⊥ ∩ b⊥ )⊥ = (a ∪ b)
17	16	lan 70	. . . . . . . . . . 11 (a⊥ ∩ (a⊥ ∩ b⊥ )⊥ ) = (a⊥ ∩ (a ∪ b))
18	17	ax-r4 36	. . . . . . . . . 10 (a⊥ ∩ (a⊥ ∩ b⊥ )⊥ )⊥ = (a⊥ ∩ (a ∪ b))⊥
19	14, 18	ax-r2 35	. . . . . . . . 9 (a ∪ (a⊥ ∩ b⊥ )) = (a⊥ ∩ (a ∪ b))⊥
20	19	con2 64	. . . . . . . 8 (a ∪ (a⊥ ∩ b⊥ ))⊥ = (a⊥ ∩ (a ∪ b))
21	20	ax-r5 37	. . . . . . 7 ((a ∪ (a⊥ ∩ b⊥ ))⊥ ∪ a) = ((a⊥ ∩ (a ∪ b)) ∪ a)
22		ax-a2 30	. . . . . . . 8 ((a⊥ ∩ (a ∪ b)) ∪ a) = (a ∪ (a⊥ ∩ (a ∪ b)))
23		oml 427	. . . . . . . 8 (a ∪ (a⊥ ∩ (a ∪ b))) = (a ∪ b)
24	22, 23	ax-r2 35	. . . . . . 7 ((a⊥ ∩ (a ∪ b)) ∪ a) = (a ∪ b)
25	21, 24	ax-r2 35	. . . . . 6 ((a ∪ (a⊥ ∩ b⊥ ))⊥ ∪ a) = (a ∪ b)
26	13, 25	ax-r2 35	. . . . 5 (((a ∪ (a⊥ ∩ b⊥ ))⊥ ∪ a) ∩ 1) = (a ∪ b)
27	12, 26	ax-r2 35	. . . 4 (((a ∪ (a⊥ ∩ b⊥ ))⊥ ∪ a) ∩ ((a ∪ (a⊥ ∩ b⊥ ))⊥ ∪ (a ∪ (a⊥ ∩ b⊥ )))) = (a ∪ b)
28	7, 27	ax-r2 35	. . 3 (((a ∪ (a⊥ ∩ b⊥ ))⊥ ∪ (a ∪ (a⊥ ∩ b⊥ ))) ∩ ((a ∪ (a⊥ ∩ b⊥ ))⊥ ∪ a)) = (a ∪ b)
29	6, 28	ax-r2 35	. 2 ((a ∪ (a⊥ ∩ b⊥ ))⊥ ∪ ((a ∪ (a⊥ ∩ b⊥ )) ∩ a)) = (a ∪ b)
30	1, 29	ax-r2 35	1 ((a ∪ (a⊥ ∩ b⊥ )) →1 a) = (a ∪ b)

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 9   →₁ wi1 13

This theorem is referenced by:  ud1 577

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a4 32  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37  ax-r3 421

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-i1 43  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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