Theorem wdid0id1 1112

Description: Show a quantum identity that follows from classical identity in a WDOL. (Contributed by NM, 5-Mar-2006.)

Hypothesis

Ref	Expression
wdid0id5.1	(a ≡0 b) = 1

Assertion

Ref	Expression
wdid0id1	(a ≡1 b) = 1

Proof of Theorem wdid0id1

Step	Hyp	Ref	Expression
1		df-id1 50	. 2 (a ≡1 b) = ((a ∪ b⊥ ) ∩ (a⊥ ∪ (a ∩ b)))
2		df-id0 49	. . . . 5 (a ≡0 b) = ((a⊥ ∪ b) ∩ (b⊥ ∪ a))
3	2	ax-r1 35	. . . 4 ((a⊥ ∪ b) ∩ (b⊥ ∪ a)) = (a ≡0 b)
4		wdid0id5.1	. . . 4 (a ≡0 b) = 1
5	3, 4	ax-r2 36	. . 3 ((a⊥ ∪ b) ∩ (b⊥ ∪ a)) = 1
6		wancom 203	. . . . . . . 8 (((a⊥ ∪ a) ∩ (a⊥ ∪ b)) ≡ ((a⊥ ∪ b) ∩ (a⊥ ∪ a))) = 1
7		wa2 192	. . . . . . . . . 10 ((a⊥ ∪ a) ≡ (a ∪ a⊥ )) = 1
8	7	wlan 370	. . . . . . . . 9 (((a⊥ ∪ b) ∩ (a⊥ ∪ a)) ≡ ((a⊥ ∪ b) ∩ (a ∪ a⊥ ))) = 1
9		wa4 194	. . . . . . . . . 10 (((a⊥ ∪ b) ∪ (a ∪ a⊥ )) ≡ (a ∪ a⊥ )) = 1
10	9	wleoa 376	. . . . . . . . 9 (((a⊥ ∪ b) ∩ (a ∪ a⊥ )) ≡ (a⊥ ∪ b)) = 1
11	8, 10	wr2 371	. . . . . . . 8 (((a⊥ ∪ b) ∩ (a⊥ ∪ a)) ≡ (a⊥ ∪ b)) = 1
12	6, 11	wr2 371	. . . . . . 7 (((a⊥ ∪ a) ∩ (a⊥ ∪ b)) ≡ (a⊥ ∪ b)) = 1
13	12	wr1 197	. . . . . 6 ((a⊥ ∪ b) ≡ ((a⊥ ∪ a) ∩ (a⊥ ∪ b))) = 1
14		wddi3 1109	. . . . . . 7 ((a⊥ ∪ (a ∩ b)) ≡ ((a⊥ ∪ a) ∩ (a⊥ ∪ b))) = 1
15	14	wr1 197	. . . . . 6 (((a⊥ ∪ a) ∩ (a⊥ ∪ b)) ≡ (a⊥ ∪ (a ∩ b))) = 1
16	13, 15	wr2 371	. . . . 5 ((a⊥ ∪ b) ≡ (a⊥ ∪ (a ∩ b))) = 1
17		wa2 192	. . . . 5 ((b⊥ ∪ a) ≡ (a ∪ b⊥ )) = 1
18	16, 17	w2an 373	. . . 4 (((a⊥ ∪ b) ∩ (b⊥ ∪ a)) ≡ ((a⊥ ∪ (a ∩ b)) ∩ (a ∪ b⊥ ))) = 1
19		biid 116	. . . 4 (((a⊥ ∪ b) ∩ (b⊥ ∪ a)) ≡ ((a⊥ ∪ b) ∩ (b⊥ ∪ a))) = 1
20		wancom 203	. . . 4 (((a ∪ b⊥ ) ∩ (a⊥ ∪ (a ∩ b))) ≡ ((a⊥ ∪ (a ∩ b)) ∩ (a ∪ b⊥ ))) = 1
21	18, 19, 20	w3tr1 374	. . 3 (((a⊥ ∪ b) ∩ (b⊥ ∪ a)) ≡ ((a ∪ b⊥ ) ∩ (a⊥ ∪ (a ∩ b)))) = 1
22	5, 21	wwbmp 205	. 2 ((a ∪ b⊥ ) ∩ (a⊥ ∪ (a ∩ b))) = 1
23	1, 22	ax-r2 36	1 (a ≡1 b) = 1

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   ≡₀ wid0 17   ≡₁ wid1 18

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-wom 361  ax-wdol 1104

This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-i2 45  df-id0 49  df-id1 50  df-le 129  df-le1 130  df-le2 131  df-cmtr 134

This theorem is referenced by:  wddi-1  1118