Theorem u5lembi 725

Description: Relevance implication and biconditional. (Contributed by NM, 17-Jan-1998.)

Assertion

Ref	Expression
u5lembi	((a →5 b) ∩ (b →5 a)) = (a ≡ b)

Proof of Theorem u5lembi

Step	Hyp	Ref	Expression
1		u5lemc1b 685	. . . . . . 7 b C (a →5 b)
2	1	comcom 453	. . . . . 6 (a →5 b) C b
3		u5lemc1 684	. . . . . . 7 a C (a →5 b)
4	3	comcom 453	. . . . . 6 (a →5 b) C a
5	2, 4	com2an 484	. . . . 5 (a →5 b) C (b ∩ a)
6	2	comcom2 183	. . . . . 6 (a →5 b) C b⊥
7	6, 4	com2an 484	. . . . 5 (a →5 b) C (b⊥ ∩ a)
8	5, 7	com2or 483	. . . 4 (a →5 b) C ((b ∩ a) ∪ (b⊥ ∩ a))
9	4	comcom2 183	. . . . 5 (a →5 b) C a⊥
10	6, 9	com2an 484	. . . 4 (a →5 b) C (b⊥ ∩ a⊥ )
11	8, 10	fh1 469	. . 3 ((a →5 b) ∩ (((b ∩ a) ∪ (b⊥ ∩ a)) ∪ (b⊥ ∩ a⊥ ))) = (((a →5 b) ∩ ((b ∩ a) ∪ (b⊥ ∩ a))) ∪ ((a →5 b) ∩ (b⊥ ∩ a⊥ )))
12	5, 7	fh1 469	. . . . . 6 ((a →5 b) ∩ ((b ∩ a) ∪ (b⊥ ∩ a))) = (((a →5 b) ∩ (b ∩ a)) ∪ ((a →5 b) ∩ (b⊥ ∩ a)))
13		ancom 74	. . . . . . . . 9 ((a →5 b) ∩ (b ∩ a)) = ((b ∩ a) ∩ (a →5 b))
14		ancom 74	. . . . . . . . . . 11 (b ∩ a) = (a ∩ b)
15		df-i5 48	. . . . . . . . . . . 12 (a →5 b) = (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (a⊥ ∩ b⊥ ))
16		ax-a3 32	. . . . . . . . . . . 12 (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (a⊥ ∩ b⊥ )) = ((a ∩ b) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))
17	15, 16	ax-r2 36	. . . . . . . . . . 11 (a →5 b) = ((a ∩ b) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))
18	14, 17	2an 79	. . . . . . . . . 10 ((b ∩ a) ∩ (a →5 b)) = ((a ∩ b) ∩ ((a ∩ b) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))))
19		anabs 121	. . . . . . . . . 10 ((a ∩ b) ∩ ((a ∩ b) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))) = (a ∩ b)
20	18, 19	ax-r2 36	. . . . . . . . 9 ((b ∩ a) ∩ (a →5 b)) = (a ∩ b)
21	13, 20	ax-r2 36	. . . . . . . 8 ((a →5 b) ∩ (b ∩ a)) = (a ∩ b)
22		anandi 114	. . . . . . . . 9 ((a →5 b) ∩ (b⊥ ∩ a)) = (((a →5 b) ∩ b⊥ ) ∩ ((a →5 b) ∩ a))
23		u5lemanb 619	. . . . . . . . . . 11 ((a →5 b) ∩ b⊥ ) = (a⊥ ∩ b⊥ )
24		u5lemaa 604	. . . . . . . . . . 11 ((a →5 b) ∩ a) = (a ∩ b)
25	23, 24	2an 79	. . . . . . . . . 10 (((a →5 b) ∩ b⊥ ) ∩ ((a →5 b) ∩ a)) = ((a⊥ ∩ b⊥ ) ∩ (a ∩ b))
26		ancom 74	. . . . . . . . . . 11 ((a⊥ ∩ b⊥ ) ∩ (a ∩ b)) = ((a ∩ b) ∩ (a⊥ ∩ b⊥ ))
27		an4 86	. . . . . . . . . . . 12 ((a ∩ b) ∩ (a⊥ ∩ b⊥ )) = ((a ∩ a⊥ ) ∩ (b ∩ b⊥ ))
28		dff 101	. . . . . . . . . . . . . . 15 0 = (b ∩ b⊥ )
29	28	ax-r1 35	. . . . . . . . . . . . . 14 (b ∩ b⊥ ) = 0
30	29	lan 77	. . . . . . . . . . . . 13 ((a ∩ a⊥ ) ∩ (b ∩ b⊥ )) = ((a ∩ a⊥ ) ∩ 0)
31		an0 108	. . . . . . . . . . . . 13 ((a ∩ a⊥ ) ∩ 0) = 0
32	30, 31	ax-r2 36	. . . . . . . . . . . 12 ((a ∩ a⊥ ) ∩ (b ∩ b⊥ )) = 0
33	27, 32	ax-r2 36	. . . . . . . . . . 11 ((a ∩ b) ∩ (a⊥ ∩ b⊥ )) = 0
34	26, 33	ax-r2 36	. . . . . . . . . 10 ((a⊥ ∩ b⊥ ) ∩ (a ∩ b)) = 0
35	25, 34	ax-r2 36	. . . . . . . . 9 (((a →5 b) ∩ b⊥ ) ∩ ((a →5 b) ∩ a)) = 0
36	22, 35	ax-r2 36	. . . . . . . 8 ((a →5 b) ∩ (b⊥ ∩ a)) = 0
37	21, 36	2or 72	. . . . . . 7 (((a →5 b) ∩ (b ∩ a)) ∪ ((a →5 b) ∩ (b⊥ ∩ a))) = ((a ∩ b) ∪ 0)
38		or0 102	. . . . . . 7 ((a ∩ b) ∪ 0) = (a ∩ b)
39	37, 38	ax-r2 36	. . . . . 6 (((a →5 b) ∩ (b ∩ a)) ∪ ((a →5 b) ∩ (b⊥ ∩ a))) = (a ∩ b)
40	12, 39	ax-r2 36	. . . . 5 ((a →5 b) ∩ ((b ∩ a) ∪ (b⊥ ∩ a))) = (a ∩ b)
41		ancom 74	. . . . . 6 ((a →5 b) ∩ (b⊥ ∩ a⊥ )) = ((b⊥ ∩ a⊥ ) ∩ (a →5 b))
42		ancom 74	. . . . . . . 8 (b⊥ ∩ a⊥ ) = (a⊥ ∩ b⊥ )
43		ax-a2 31	. . . . . . . . 9 (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (a⊥ ∩ b⊥ )) = ((a⊥ ∩ b⊥ ) ∪ ((a ∩ b) ∪ (a⊥ ∩ b)))
44	15, 43	ax-r2 36	. . . . . . . 8 (a →5 b) = ((a⊥ ∩ b⊥ ) ∪ ((a ∩ b) ∪ (a⊥ ∩ b)))
45	42, 44	2an 79	. . . . . . 7 ((b⊥ ∩ a⊥ ) ∩ (a →5 b)) = ((a⊥ ∩ b⊥ ) ∩ ((a⊥ ∩ b⊥ ) ∪ ((a ∩ b) ∪ (a⊥ ∩ b))))
46		anabs 121	. . . . . . 7 ((a⊥ ∩ b⊥ ) ∩ ((a⊥ ∩ b⊥ ) ∪ ((a ∩ b) ∪ (a⊥ ∩ b)))) = (a⊥ ∩ b⊥ )
47	45, 46	ax-r2 36	. . . . . 6 ((b⊥ ∩ a⊥ ) ∩ (a →5 b)) = (a⊥ ∩ b⊥ )
48	41, 47	ax-r2 36	. . . . 5 ((a →5 b) ∩ (b⊥ ∩ a⊥ )) = (a⊥ ∩ b⊥ )
49	40, 48	2or 72	. . . 4 (((a →5 b) ∩ ((b ∩ a) ∪ (b⊥ ∩ a))) ∪ ((a →5 b) ∩ (b⊥ ∩ a⊥ ))) = ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))
50		id 59	. . . 4 ((a ∩ b) ∪ (a⊥ ∩ b⊥ )) = ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))
51	49, 50	ax-r2 36	. . 3 (((a →5 b) ∩ ((b ∩ a) ∪ (b⊥ ∩ a))) ∪ ((a →5 b) ∩ (b⊥ ∩ a⊥ ))) = ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))
52	11, 51	ax-r2 36	. 2 ((a →5 b) ∩ (((b ∩ a) ∪ (b⊥ ∩ a)) ∪ (b⊥ ∩ a⊥ ))) = ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))
53		df-i5 48	. . 3 (b →5 a) = (((b ∩ a) ∪ (b⊥ ∩ a)) ∪ (b⊥ ∩ a⊥ ))
54	53	lan 77	. 2 ((a →5 b) ∩ (b →5 a)) = ((a →5 b) ∩ (((b ∩ a) ∪ (b⊥ ∩ a)) ∪ (b⊥ ∩ a⊥ )))
55		dfb 94	. 2 (a ≡ b) = ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))
56	52, 54, 55	3tr1 63	1 ((a →5 b) ∩ (b →5 a)) = (a ≡ b)

Syntax hints:   = wb 1  ^⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7  0wf 9   →₅ wi5 16

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-r3 439

This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i5 48  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by:  oago3.21x  890