Theorem 2vwomlem 347

Description: Lemma from 2-variable WOML rule.

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 86	. 2 (a == b) = ((a ^ b) v (a_|_ ^ b_|_))
2		df-f 41	. . . . 5 0 = 1_|_
3		anor2 81	. . . . . . 7 (a_|_ ^ (a v b)) = (a v (a v b)_|_)_|_
4	3	ax-r1 34	. . . . . 6 (a v (a v b)_|_)_|_ = (a_|_ ^ (a v b))
5		anor3 82	. . . . . . . . . . 11 (a_|_ ^ b_|_) = (a v b)_|_
6	5	ax-r1 34	. . . . . . . . . 10 (a v b)_|_ = (a_|_ ^ b_|_)
7		ancom 68	. . . . . . . . . 10 (a_|_ ^ b_|_) = (b_|_ ^ a_|_)
8	6, 7	ax-r2 35	. . . . . . . . 9 (a v b)_|_ = (b_|_ ^ a_|_)
9	8	lor 66	. . . . . . . 8 (a v (a v b)_|_) = (a v (b_|_ ^ a_|_))
10		df-i2 44	. . . . . . . . 9 (b ->2 a) = (a v (b_|_ ^ a_|_))
11	10	ax-r1 34	. . . . . . . 8 (a v (b_|_ ^ a_|_)) = (b ->2 a)
12		2vwomlem.2	. . . . . . . 8 (b ->2 a) = 1
13	9, 11, 12	3tr 62	. . . . . . 7 (a v (a v b)_|_) = 1
14	13	ax-r4 36	. . . . . 6 (a v (a v b)_|_)_|_ = 1_|_
15		a5c 113	. . . . . . . . 9 (a_|_ ^ (a_|_ v b_|_)) = a_|_
16	15	ax-r1 34	. . . . . . . 8 a_|_ = (a_|_ ^ (a_|_ v b_|_))
17	16	ran 71	. . . . . . 7 (a_|_ ^ (a v b)) = ((a_|_ ^ (a_|_ v b_|_)) ^ (a v b))
18		anass 69	. . . . . . 7 ((a_|_ ^ (a_|_ v b_|_)) ^ (a v b)) = (a_|_ ^ ((a_|_ v b_|_) ^ (a v b)))
19		oran3 85	. . . . . . . . . 10 (a_|_ v b_|_) = (a ^ b)_|_
20		oran 79	. . . . . . . . . 10 (a v b) = (a_|_ ^ b_|_)_|_
21	19, 20	2an 72	. . . . . . . . 9 ((a_|_ v b_|_) ^ (a v b)) = ((a ^ b)_|_ ^ (a_|_ ^ b_|_)_|_)
22		anor3 82	. . . . . . . . 9 ((a ^ b)_|_ ^ (a_|_ ^ b_|_)_|_) = ((a ^ b) v (a_|_ ^ b_|_))_|_
23	21, 22	ax-r2 35	. . . . . . . 8 ((a_|_ v b_|_) ^ (a v b)) = ((a ^ b) v (a_|_ ^ b_|_))_|_
24	23	lan 70	. . . . . . 7 (a_|_ ^ ((a_|_ v b_|_) ^ (a v b))) = (a_|_ ^ ((a ^ b) v (a_|_ ^ b_|_))_|_)
25	17, 18, 24	3tr 62	. . . . . 6 (a_|_ ^ (a v b)) = (a_|_ ^ ((a ^ b) v (a_|_ ^ b_|_))_|_)
26	4, 14, 25	3tr2 61	. . . . 5 1_|_ = (a_|_ ^ ((a ^ b) v (a_|_ ^ b_|_))_|_)
27	2, 26	ax-r2 35	. . . 4 0 = (a_|_ ^ ((a ^ b) v (a_|_ ^ b_|_))_|_)
28	27	lor 66	. . 3 (((a ^ b) v (a_|_ ^ b_|_)) v 0) = (((a ^ b) v (a_|_ ^ b_|_)) v (a_|_ ^ ((a ^ b) v (a_|_ ^ b_|_))_|_))
29		or0 94	. . 3 (((a ^ b) v (a_|_ ^ b_|_)) v 0) = ((a ^ b) v (a_|_ ^ b_|_))
30		le1 138	. . . . 5 (a_|_ v (a ^ ((a ^ b) v (a_|_ ^ b_|_)))) =< 1
31		df-i2 44	. . . . . . . . . 10 (a ->2 b) = (b v (a_|_ ^ b_|_))
32	31	ax-r1 34	. . . . . . . . 9 (b v (a_|_ ^ b_|_)) = (a ->2 b)
33		2vwomlem.1	. . . . . . . . 9 (a ->2 b) = 1
34	32, 33	ax-r2 35	. . . . . . . 8 (b v (a_|_ ^ b_|_)) = 1
35	34	2vwomr2 344	. . . . . . 7 (a_|_ v (a ^ b)) = 1
36	35	ax-r1 34	. . . . . 6 1 = (a_|_ v (a ^ b))
37		lea 152	. . . . . . . 8 (a ^ b) =< a
38		leo 150	. . . . . . . 8 (a ^ b) =< ((a ^ b) v (a_|_ ^ b_|_))
39	37, 38	ler2an 165	. . . . . . 7 (a ^ b) =< (a ^ ((a ^ b) v (a_|_ ^ b_|_)))
40	39	lelor 158	. . . . . 6 (a_|_ v (a ^ b)) =< (a_|_ v (a ^ ((a ^ b) v (a_|_ ^ b_|_))))
41	36, 40	bltr 130	. . . . 5 1 =< (a_|_ v (a ^ ((a ^ b) v (a_|_ ^ b_|_))))
42	30, 41	lebi 137	. . . 4 (a_|_ v (a ^ ((a ^ b) v (a_|_ ^ b_|_)))) = 1
43	42	ax-wom 343	. . 3 (((a ^ b) v (a_|_ ^ b_|_)) v (a_|_ ^ ((a ^ b) v (a_|_ ^ b_|_))_|_)) = 1
44	28, 29, 43	3tr2 61	. 2 ((a ^ b) v (a_|_ ^ b_|_)) = 1
45	1, 44	ax-r2 35	1 (a == b) = 1

Syntax hints:   = wb 1  _|_wn 4   == tb 5   v wo 6   ^ wa 7  1wt 9  0wf 10   ->2 wi2 14

This theorem is referenced by:  wr5-2v 348

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-wom 343

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-i2 44  df-le1 122  df-le2 123

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