Theorem bi1o1a 780

Description: Equivalence to biconditional.

Assertion

Ref	Expression
bi1o1a	(a ≡ b) = ((a →1 (a ∩ b)) ∩ ((a ∪ b) →1 a))

Proof of Theorem bi1o1a

Step	Hyp	Ref	Expression
1		lea 152	. . . . . . 7 (a⊥ ∩ b⊥ ) ≤ a⊥
2		leo 150	. . . . . . 7 a⊥ ≤ (a⊥ ∪ (a ∩ b))
3	1, 2	letr 129	. . . . . 6 (a⊥ ∩ b⊥ ) ≤ (a⊥ ∪ (a ∩ b))
4	3	lecom 172	. . . . 5 (a⊥ ∩ b⊥ ) C (a⊥ ∪ (a ∩ b))
5	4	comcom 435	. . . 4 (a⊥ ∪ (a ∩ b)) C (a⊥ ∩ b⊥ )
6		comor1 443	. . . . 5 (a⊥ ∪ (a ∩ b)) C a⊥
7	6	comcom7 442	. . . 4 (a⊥ ∪ (a ∩ b)) C a
8	5, 7	fh1 451	. . 3 ((a⊥ ∪ (a ∩ b)) ∩ ((a⊥ ∩ b⊥ ) ∪ a)) = (((a⊥ ∪ (a ∩ b)) ∩ (a⊥ ∩ b⊥ )) ∪ ((a⊥ ∪ (a ∩ b)) ∩ a))
9	8	ax-r1 34	. 2 (((a⊥ ∪ (a ∩ b)) ∩ (a⊥ ∩ b⊥ )) ∪ ((a⊥ ∪ (a ∩ b)) ∩ a)) = ((a⊥ ∪ (a ∩ b)) ∩ ((a⊥ ∩ b⊥ ) ∪ a))
10		dfb 86	. . 3 (a ≡ b) = ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))
11		ax-a2 30	. . 3 ((a ∩ b) ∪ (a⊥ ∩ b⊥ )) = ((a⊥ ∩ b⊥ ) ∪ (a ∩ b))
12		leid 140	. . . . . 6 (a⊥ ∩ b⊥ ) ≤ (a⊥ ∩ b⊥ )
13	3, 12	ler2an 165	. . . . 5 (a⊥ ∩ b⊥ ) ≤ ((a⊥ ∪ (a ∩ b)) ∩ (a⊥ ∩ b⊥ ))
14		lear 153	. . . . 5 ((a⊥ ∪ (a ∩ b)) ∩ (a⊥ ∩ b⊥ )) ≤ (a⊥ ∩ b⊥ )
15	13, 14	lebi 137	. . . 4 (a⊥ ∩ b⊥ ) = ((a⊥ ∪ (a ∩ b)) ∩ (a⊥ ∩ b⊥ ))
16		dff 93	. . . . . . 7 0 = (a ∩ a⊥ )
17		ancom 68	. . . . . . 7 (a ∩ a⊥ ) = (a⊥ ∩ a)
18	16, 17	ax-r2 35	. . . . . 6 0 = (a⊥ ∩ a)
19	18	ax-r5 37	. . . . 5 (0 ∪ ((a ∩ b) ∩ a)) = ((a⊥ ∩ a) ∪ ((a ∩ b) ∩ a))
20		lea 152	. . . . . . . 8 (a ∩ b) ≤ a
21	20	df2le2 128	. . . . . . 7 ((a ∩ b) ∩ a) = (a ∩ b)
22	21	ax-r1 34	. . . . . 6 (a ∩ b) = ((a ∩ b) ∩ a)
23		or0r 95	. . . . . . 7 (0 ∪ ((a ∩ b) ∩ a)) = ((a ∩ b) ∩ a)
24	23	ax-r1 34	. . . . . 6 ((a ∩ b) ∩ a) = (0 ∪ ((a ∩ b) ∩ a))
25	22, 24	ax-r2 35	. . . . 5 (a ∩ b) = (0 ∪ ((a ∩ b) ∩ a))
26		comid 179	. . . . . . 7 a C a
27	26	comcom2 175	. . . . . 6 a C a⊥
28		comanr1 446	. . . . . 6 a C (a ∩ b)
29	27, 28	fh1r 455	. . . . 5 ((a⊥ ∪ (a ∩ b)) ∩ a) = ((a⊥ ∩ a) ∪ ((a ∩ b) ∩ a))
30	19, 25, 29	3tr1 60	. . . 4 (a ∩ b) = ((a⊥ ∪ (a ∩ b)) ∩ a)
31	15, 30	2or 67	. . 3 ((a⊥ ∩ b⊥ ) ∪ (a ∩ b)) = (((a⊥ ∪ (a ∩ b)) ∩ (a⊥ ∩ b⊥ )) ∪ ((a⊥ ∪ (a ∩ b)) ∩ a))
32	10, 11, 31	3tr 62	. 2 (a ≡ b) = (((a⊥ ∪ (a ∩ b)) ∩ (a⊥ ∩ b⊥ )) ∪ ((a⊥ ∪ (a ∩ b)) ∩ a))
33		df-i1 43	. . . 4 (a →1 (a ∩ b)) = (a⊥ ∪ (a ∩ (a ∩ b)))
34		lear 153	. . . . . 6 (a ∩ (a ∩ b)) ≤ (a ∩ b)
35		leid 140	. . . . . . 7 (a ∩ b) ≤ (a ∩ b)
36	20, 35	ler2an 165	. . . . . 6 (a ∩ b) ≤ (a ∩ (a ∩ b))
37	34, 36	lebi 137	. . . . 5 (a ∩ (a ∩ b)) = (a ∩ b)
38	37	lor 66	. . . 4 (a⊥ ∪ (a ∩ (a ∩ b))) = (a⊥ ∪ (a ∩ b))
39	33, 38	ax-r2 35	. . 3 (a →1 (a ∩ b)) = (a⊥ ∪ (a ∩ b))
40		df-i1 43	. . . 4 ((a ∪ b) →1 a) = ((a ∪ b)⊥ ∪ ((a ∪ b) ∩ a))
41		anor3 82	. . . . . 6 (a⊥ ∩ b⊥ ) = (a ∪ b)⊥
42	41	ax-r1 34	. . . . 5 (a ∪ b)⊥ = (a⊥ ∩ b⊥ )
43		lear 153	. . . . . 6 ((a ∪ b) ∩ a) ≤ a
44		leo 150	. . . . . . 7 a ≤ (a ∪ b)
45		leid 140	. . . . . . 7 a ≤ a
46	44, 45	ler2an 165	. . . . . 6 a ≤ ((a ∪ b) ∩ a)
47	43, 46	lebi 137	. . . . 5 ((a ∪ b) ∩ a) = a
48	42, 47	2or 67	. . . 4 ((a ∪ b)⊥ ∪ ((a ∪ b) ∩ a)) = ((a⊥ ∩ b⊥ ) ∪ a)
49	40, 48	ax-r2 35	. . 3 ((a ∪ b) →1 a) = ((a⊥ ∩ b⊥ ) ∪ a)
50	39, 49	2an 72	. 2 ((a →1 (a ∩ b)) ∩ ((a ∪ b) →1 a)) = ((a⊥ ∪ (a ∩ b)) ∩ ((a⊥ ∩ b⊥ ) ∪ a))
51	9, 32, 50	3tr1 60	1 (a ≡ b) = ((a →1 (a ∩ b)) ∩ ((a ∪ b) →1 a))

Syntax hints:   = wb 1  ^⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7  0wf 10   →₁ wi1 13

This theorem is referenced by:  mlaconj 827

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-i1 43  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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