Theorem u5lembi 707

Description: Relevance implication and biconditional.

Assertion

Ref	Expression
u5lembi	((a ->5 b) ^ (b ->5 a)) = (a == b)

Proof of Theorem u5lembi

Step	Hyp	Ref	Expression
1		u5lemc1b 667	. . . . . . 7 b C (a ->5 b)
2	1	comcom 435	. . . . . 6 (a ->5 b) C b
3		u5lemc1 666	. . . . . . 7 a C (a ->5 b)
4	3	comcom 435	. . . . . 6 (a ->5 b) C a
5	2, 4	com2an 466	. . . . 5 (a ->5 b) C (b ^ a)
6	2	comcom2 175	. . . . . 6 (a ->5 b) C b_|_
7	6, 4	com2an 466	. . . . 5 (a ->5 b) C (b_|_ ^ a)
8	5, 7	com2or 465	. . . 4 (a ->5 b) C ((b ^ a) v (b_|_ ^ a))
9	4	comcom2 175	. . . . 5 (a ->5 b) C a_|_
10	6, 9	com2an 466	. . . 4 (a ->5 b) C (b_|_ ^ a_|_)
11	8, 10	fh1 451	. . 3 ((a ->5 b) ^ (((b ^ a) v (b_|_ ^ a)) v (b_|_ ^ a_|_))) = (((a ->5 b) ^ ((b ^ a) v (b_|_ ^ a))) v ((a ->5 b) ^ (b_|_ ^ a_|_)))
12	5, 7	fh1 451	. . . . . 6 ((a ->5 b) ^ ((b ^ a) v (b_|_ ^ a))) = (((a ->5 b) ^ (b ^ a)) v ((a ->5 b) ^ (b_|_ ^ a)))
13		ancom 68	. . . . . . . . 9 ((a ->5 b) ^ (b ^ a)) = ((b ^ a) ^ (a ->5 b))
14		ancom 68	. . . . . . . . . . 11 (b ^ a) = (a ^ b)
15		df-i5 47	. . . . . . . . . . . 12 (a ->5 b) = (((a ^ b) v (a_|_ ^ b)) v (a_|_ ^ b_|_))
16		ax-a3 31	. . . . . . . . . . . 12 (((a ^ b) v (a_|_ ^ b)) v (a_|_ ^ b_|_)) = ((a ^ b) v ((a_|_ ^ b) v (a_|_ ^ b_|_)))
17	15, 16	ax-r2 35	. . . . . . . . . . 11 (a ->5 b) = ((a ^ b) v ((a_|_ ^ b) v (a_|_ ^ b_|_)))
18	14, 17	2an 72	. . . . . . . . . 10 ((b ^ a) ^ (a ->5 b)) = ((a ^ b) ^ ((a ^ b) v ((a_|_ ^ b) v (a_|_ ^ b_|_))))
19		a5c 113	. . . . . . . . . 10 ((a ^ b) ^ ((a ^ b) v ((a_|_ ^ b) v (a_|_ ^ b_|_)))) = (a ^ b)
20	18, 19	ax-r2 35	. . . . . . . . 9 ((b ^ a) ^ (a ->5 b)) = (a ^ b)
21	13, 20	ax-r2 35	. . . . . . . 8 ((a ->5 b) ^ (b ^ a)) = (a ^ b)
22		anandi 106	. . . . . . . . 9 ((a ->5 b) ^ (b_|_ ^ a)) = (((a ->5 b) ^ b_|_) ^ ((a ->5 b) ^ a))
23		u5lemanb 601	. . . . . . . . . . 11 ((a ->5 b) ^ b_|_) = (a_|_ ^ b_|_)
24		u5lemaa 586	. . . . . . . . . . 11 ((a ->5 b) ^ a) = (a ^ b)
25	23, 24	2an 72	. . . . . . . . . 10 (((a ->5 b) ^ b_|_) ^ ((a ->5 b) ^ a)) = ((a_|_ ^ b_|_) ^ (a ^ b))
26		ancom 68	. . . . . . . . . . 11 ((a_|_ ^ b_|_) ^ (a ^ b)) = ((a ^ b) ^ (a_|_ ^ b_|_))
27		an4 78	. . . . . . . . . . . 12 ((a ^ b) ^ (a_|_ ^ b_|_)) = ((a ^ a_|_) ^ (b ^ b_|_))
28		dff 93	. . . . . . . . . . . . . . 15 0 = (b ^ b_|_)
29	28	ax-r1 34	. . . . . . . . . . . . . 14 (b ^ b_|_) = 0
30	29	lan 70	. . . . . . . . . . . . 13 ((a ^ a_|_) ^ (b ^ b_|_)) = ((a ^ a_|_) ^ 0)
31		an0 100	. . . . . . . . . . . . 13 ((a ^ a_|_) ^ 0) = 0
32	30, 31	ax-r2 35	. . . . . . . . . . . 12 ((a ^ a_|_) ^ (b ^ b_|_)) = 0
33	27, 32	ax-r2 35	. . . . . . . . . . 11 ((a ^ b) ^ (a_|_ ^ b_|_)) = 0
34	26, 33	ax-r2 35	. . . . . . . . . 10 ((a_|_ ^ b_|_) ^ (a ^ b)) = 0
35	25, 34	ax-r2 35	. . . . . . . . 9 (((a ->5 b) ^ b_|_) ^ ((a ->5 b) ^ a)) = 0
36	22, 35	ax-r2 35	. . . . . . . 8 ((a ->5 b) ^ (b_|_ ^ a)) = 0
37	21, 36	2or 67	. . . . . . 7 (((a ->5 b) ^ (b ^ a)) v ((a ->5 b) ^ (b_|_ ^ a))) = ((a ^ b) v 0)
38		or0 94	. . . . . . 7 ((a ^ b) v 0) = (a ^ b)
39	37, 38	ax-r2 35	. . . . . 6 (((a ->5 b) ^ (b ^ a)) v ((a ->5 b) ^ (b_|_ ^ a))) = (a ^ b)
40	12, 39	ax-r2 35	. . . . 5 ((a ->5 b) ^ ((b ^ a) v (b_|_ ^ a))) = (a ^ b)
41		ancom 68	. . . . . 6 ((a ->5 b) ^ (b_|_ ^ a_|_)) = ((b_|_ ^ a_|_) ^ (a ->5 b))
42		ancom 68	. . . . . . . 8 (b_|_ ^ a_|_) = (a_|_ ^ b_|_)
43		ax-a2 30	. . . . . . . . 9 (((a ^ b) v (a_|_ ^ b)) v (a_|_ ^ b_|_)) = ((a_|_ ^ b_|_) v ((a ^ b) v (a_|_ ^ b)))
44	15, 43	ax-r2 35	. . . . . . . 8 (a ->5 b) = ((a_|_ ^ b_|_) v ((a ^ b) v (a_|_ ^ b)))
45	42, 44	2an 72	. . . . . . 7 ((b_|_ ^ a_|_) ^ (a ->5 b)) = ((a_|_ ^ b_|_) ^ ((a_|_ ^ b_|_) v ((a ^ b) v (a_|_ ^ b))))
46		a5c 113	. . . . . . 7 ((a_|_ ^ b_|_) ^ ((a_|_ ^ b_|_) v ((a ^ b) v (a_|_ ^ b)))) = (a_|_ ^ b_|_)
47	45, 46	ax-r2 35	. . . . . 6 ((b_|_ ^ a_|_) ^ (a ->5 b)) = (a_|_ ^ b_|_)
48	41, 47	ax-r2 35	. . . . 5 ((a ->5 b) ^ (b_|_ ^ a_|_)) = (a_|_ ^ b_|_)
49	40, 48	2or 67	. . . 4 (((a ->5 b) ^ ((b ^ a) v (b_|_ ^ a))) v ((a ->5 b) ^ (b_|_ ^ a_|_))) = ((a ^ b) v (a_|_ ^ b_|_))
50		id 58	. . . 4 ((a ^ b) v (a_|_ ^ b_|_)) = ((a ^ b) v (a_|_ ^ b_|_))
51	49, 50	ax-r2 35	. . 3 (((a ->5 b) ^ ((b ^ a) v (b_|_ ^ a))) v ((a ->5 b) ^ (b_|_ ^ a_|_))) = ((a ^ b) v (a_|_ ^ b_|_))
52	11, 51	ax-r2 35	. 2 ((a ->5 b) ^ (((b ^ a) v (b_|_ ^ a)) v (b_|_ ^ a_|_))) = ((a ^ b) v (a_|_ ^ b_|_))
53		df-i5 47	. . 3 (b ->5 a) = (((b ^ a) v (b_|_ ^ a)) v (b_|_ ^ a_|_))
54	53	lan 70	. 2 ((a ->5 b) ^ (b ->5 a)) = ((a ->5 b) ^ (((b ^ a) v (b_|_ ^ a)) v (b_|_ ^ a_|_)))
55		dfb 86	. 2 (a == b) = ((a ^ b) v (a_|_ ^ b_|_))
56	52, 54, 55	3tr1 60	1 ((a ->5 b) ^ (b ->5 a)) = (a == b)

Syntax hints:   = wb 1  _|_wn 4   == tb 5   v wo 6   ^ wa 7  0wf 10   ->5 wi5 17

This theorem is referenced by:  oago3.21x 872

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-i5 47  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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