Theorem 2vwomlem 365

Description: Lemma from 2-variable WOML rule. (Contributed by NM, 13-Nov-1998.)

Hypotheses

Ref	Expression
2vwomlem.1	(a →2 b) = 1
2vwomlem.2	(b →2 a) = 1

Assertion

Ref	Expression
2vwomlem	(a ≡ b) = 1

Proof of Theorem 2vwomlem

Step	Hyp	Ref	Expression
1		dfb 94	. 2 (a ≡ b) = ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))
2		df-f 42	. . . . 5 0 = 1⊥
3		anor2 89	. . . . . . 7 (a⊥ ∩ (a ∪ b)) = (a ∪ (a ∪ b)⊥ )⊥
4	3	ax-r1 35	. . . . . 6 (a ∪ (a ∪ b)⊥ )⊥ = (a⊥ ∩ (a ∪ b))
5		anor3 90	. . . . . . . . . . 11 (a⊥ ∩ b⊥ ) = (a ∪ b)⊥
6	5	ax-r1 35	. . . . . . . . . 10 (a ∪ b)⊥ = (a⊥ ∩ b⊥ )
7		ancom 74	. . . . . . . . . 10 (a⊥ ∩ b⊥ ) = (b⊥ ∩ a⊥ )
8	6, 7	ax-r2 36	. . . . . . . . 9 (a ∪ b)⊥ = (b⊥ ∩ a⊥ )
9	8	lor 70	. . . . . . . 8 (a ∪ (a ∪ b)⊥ ) = (a ∪ (b⊥ ∩ a⊥ ))
10		df-i2 45	. . . . . . . . 9 (b →2 a) = (a ∪ (b⊥ ∩ a⊥ ))
11	10	ax-r1 35	. . . . . . . 8 (a ∪ (b⊥ ∩ a⊥ )) = (b →2 a)
12		2vwomlem.2	. . . . . . . 8 (b →2 a) = 1
13	9, 11, 12	3tr 65	. . . . . . 7 (a ∪ (a ∪ b)⊥ ) = 1
14	13	ax-r4 37	. . . . . 6 (a ∪ (a ∪ b)⊥ )⊥ = 1⊥
15		anabs 121	. . . . . . . . 9 (a⊥ ∩ (a⊥ ∪ b⊥ )) = a⊥
16	15	ax-r1 35	. . . . . . . 8 a⊥ = (a⊥ ∩ (a⊥ ∪ b⊥ ))
17	16	ran 78	. . . . . . 7 (a⊥ ∩ (a ∪ b)) = ((a⊥ ∩ (a⊥ ∪ b⊥ )) ∩ (a ∪ b))
18		anass 76	. . . . . . 7 ((a⊥ ∩ (a⊥ ∪ b⊥ )) ∩ (a ∪ b)) = (a⊥ ∩ ((a⊥ ∪ b⊥ ) ∩ (a ∪ b)))
19		oran3 93	. . . . . . . . . 10 (a⊥ ∪ b⊥ ) = (a ∩ b)⊥
20		oran 87	. . . . . . . . . 10 (a ∪ b) = (a⊥ ∩ b⊥ )⊥
21	19, 20	2an 79	. . . . . . . . 9 ((a⊥ ∪ b⊥ ) ∩ (a ∪ b)) = ((a ∩ b)⊥ ∩ (a⊥ ∩ b⊥ )⊥ )
22		anor3 90	. . . . . . . . 9 ((a ∩ b)⊥ ∩ (a⊥ ∩ b⊥ )⊥ ) = ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))⊥
23	21, 22	ax-r2 36	. . . . . . . 8 ((a⊥ ∪ b⊥ ) ∩ (a ∪ b)) = ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))⊥
24	23	lan 77	. . . . . . 7 (a⊥ ∩ ((a⊥ ∪ b⊥ ) ∩ (a ∪ b))) = (a⊥ ∩ ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))⊥ )
25	17, 18, 24	3tr 65	. . . . . 6 (a⊥ ∩ (a ∪ b)) = (a⊥ ∩ ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))⊥ )
26	4, 14, 25	3tr2 64	. . . . 5 1⊥ = (a⊥ ∩ ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))⊥ )
27	2, 26	ax-r2 36	. . . 4 0 = (a⊥ ∩ ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))⊥ )
28	27	lor 70	. . 3 (((a ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ 0) = (((a ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a⊥ ∩ ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))⊥ ))
29		or0 102	. . 3 (((a ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ 0) = ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))
30		le1 146	. . . . 5 (a⊥ ∪ (a ∩ ((a ∩ b) ∪ (a⊥ ∩ b⊥ )))) ≤ 1
31		df-i2 45	. . . . . . . . . 10 (a →2 b) = (b ∪ (a⊥ ∩ b⊥ ))
32	31	ax-r1 35	. . . . . . . . 9 (b ∪ (a⊥ ∩ b⊥ )) = (a →2 b)
33		2vwomlem.1	. . . . . . . . 9 (a →2 b) = 1
34	32, 33	ax-r2 36	. . . . . . . 8 (b ∪ (a⊥ ∩ b⊥ )) = 1
35	34	2vwomr2 362	. . . . . . 7 (a⊥ ∪ (a ∩ b)) = 1
36	35	ax-r1 35	. . . . . 6 1 = (a⊥ ∪ (a ∩ b))
37		lea 160	. . . . . . . 8 (a ∩ b) ≤ a
38		leo 158	. . . . . . . 8 (a ∩ b) ≤ ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))
39	37, 38	ler2an 173	. . . . . . 7 (a ∩ b) ≤ (a ∩ ((a ∩ b) ∪ (a⊥ ∩ b⊥ )))
40	39	lelor 166	. . . . . 6 (a⊥ ∪ (a ∩ b)) ≤ (a⊥ ∪ (a ∩ ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))))
41	36, 40	bltr 138	. . . . 5 1 ≤ (a⊥ ∪ (a ∩ ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))))
42	30, 41	lebi 145	. . . 4 (a⊥ ∪ (a ∩ ((a ∩ b) ∪ (a⊥ ∩ b⊥ )))) = 1
43	42	ax-wom 361	. . 3 (((a ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a⊥ ∩ ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))⊥ )) = 1
44	28, 29, 43	3tr2 64	. 2 ((a ∩ b) ∪ (a⊥ ∩ b⊥ )) = 1
45	1, 44	ax-r2 36	1 (a ≡ b) = 1

Syntax hints:   = wb 1  ^⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7  1wt 8  0wf 9   →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-wom 361

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

This theorem is referenced by:  wr5-2v  366