Theorem bi1o1a 798

Description: Equivalence to biconditional. (Contributed by NM, 5-Jul-2000.)

Assertion

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

Proof of Theorem bi1o1a

Step	Hyp	Ref	Expression
1		lea 160	. . . . . . 7 (a⊥ ∩ b⊥ ) ≤ a⊥
2		leo 158	. . . . . . 7 a⊥ ≤ (a⊥ ∪ (a ∩ b))
3	1, 2	letr 137	. . . . . 6 (a⊥ ∩ b⊥ ) ≤ (a⊥ ∪ (a ∩ b))
4	3	lecom 180	. . . . 5 (a⊥ ∩ b⊥ ) C (a⊥ ∪ (a ∩ b))
5	4	comcom 453	. . . 4 (a⊥ ∪ (a ∩ b)) C (a⊥ ∩ b⊥ )
6		comor1 461	. . . . 5 (a⊥ ∪ (a ∩ b)) C a⊥
7	6	comcom7 460	. . . 4 (a⊥ ∪ (a ∩ b)) C a
8	5, 7	fh1 469	. . 3 ((a⊥ ∪ (a ∩ b)) ∩ ((a⊥ ∩ b⊥ ) ∪ a)) = (((a⊥ ∪ (a ∩ b)) ∩ (a⊥ ∩ b⊥ )) ∪ ((a⊥ ∪ (a ∩ b)) ∩ a))
9	8	ax-r1 35	. 2 (((a⊥ ∪ (a ∩ b)) ∩ (a⊥ ∩ b⊥ )) ∪ ((a⊥ ∪ (a ∩ b)) ∩ a)) = ((a⊥ ∪ (a ∩ b)) ∩ ((a⊥ ∩ b⊥ ) ∪ a))
10		dfb 94	. . 3 (a ≡ b) = ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))
11		ax-a2 31	. . 3 ((a ∩ b) ∪ (a⊥ ∩ b⊥ )) = ((a⊥ ∩ b⊥ ) ∪ (a ∩ b))
12		leid 148	. . . . . 6 (a⊥ ∩ b⊥ ) ≤ (a⊥ ∩ b⊥ )
13	3, 12	ler2an 173	. . . . 5 (a⊥ ∩ b⊥ ) ≤ ((a⊥ ∪ (a ∩ b)) ∩ (a⊥ ∩ b⊥ ))
14		lear 161	. . . . 5 ((a⊥ ∪ (a ∩ b)) ∩ (a⊥ ∩ b⊥ )) ≤ (a⊥ ∩ b⊥ )
15	13, 14	lebi 145	. . . 4 (a⊥ ∩ b⊥ ) = ((a⊥ ∪ (a ∩ b)) ∩ (a⊥ ∩ b⊥ ))
16		dff 101	. . . . . . 7 0 = (a ∩ a⊥ )
17		ancom 74	. . . . . . 7 (a ∩ a⊥ ) = (a⊥ ∩ a)
18	16, 17	ax-r2 36	. . . . . 6 0 = (a⊥ ∩ a)
19	18	ax-r5 38	. . . . 5 (0 ∪ ((a ∩ b) ∩ a)) = ((a⊥ ∩ a) ∪ ((a ∩ b) ∩ a))
20		lea 160	. . . . . . . 8 (a ∩ b) ≤ a
21	20	df2le2 136	. . . . . . 7 ((a ∩ b) ∩ a) = (a ∩ b)
22	21	ax-r1 35	. . . . . 6 (a ∩ b) = ((a ∩ b) ∩ a)
23		or0r 103	. . . . . . 7 (0 ∪ ((a ∩ b) ∩ a)) = ((a ∩ b) ∩ a)
24	23	ax-r1 35	. . . . . 6 ((a ∩ b) ∩ a) = (0 ∪ ((a ∩ b) ∩ a))
25	22, 24	ax-r2 36	. . . . 5 (a ∩ b) = (0 ∪ ((a ∩ b) ∩ a))
26		comid 187	. . . . . . 7 a C a
27	26	comcom2 183	. . . . . 6 a C a⊥
28		comanr1 464	. . . . . 6 a C (a ∩ b)
29	27, 28	fh1r 473	. . . . 5 ((a⊥ ∪ (a ∩ b)) ∩ a) = ((a⊥ ∩ a) ∪ ((a ∩ b) ∩ a))
30	19, 25, 29	3tr1 63	. . . 4 (a ∩ b) = ((a⊥ ∪ (a ∩ b)) ∩ a)
31	15, 30	2or 72	. . 3 ((a⊥ ∩ b⊥ ) ∪ (a ∩ b)) = (((a⊥ ∪ (a ∩ b)) ∩ (a⊥ ∩ b⊥ )) ∪ ((a⊥ ∪ (a ∩ b)) ∩ a))
32	10, 11, 31	3tr 65	. 2 (a ≡ b) = (((a⊥ ∪ (a ∩ b)) ∩ (a⊥ ∩ b⊥ )) ∪ ((a⊥ ∪ (a ∩ b)) ∩ a))
33		df-i1 44	. . . 4 (a →1 (a ∩ b)) = (a⊥ ∪ (a ∩ (a ∩ b)))
34		lear 161	. . . . . 6 (a ∩ (a ∩ b)) ≤ (a ∩ b)
35		leid 148	. . . . . . 7 (a ∩ b) ≤ (a ∩ b)
36	20, 35	ler2an 173	. . . . . 6 (a ∩ b) ≤ (a ∩ (a ∩ b))
37	34, 36	lebi 145	. . . . 5 (a ∩ (a ∩ b)) = (a ∩ b)
38	37	lor 70	. . . 4 (a⊥ ∪ (a ∩ (a ∩ b))) = (a⊥ ∪ (a ∩ b))
39	33, 38	ax-r2 36	. . 3 (a →1 (a ∩ b)) = (a⊥ ∪ (a ∩ b))
40		df-i1 44	. . . 4 ((a ∪ b) →1 a) = ((a ∪ b)⊥ ∪ ((a ∪ b) ∩ a))
41		anor3 90	. . . . . 6 (a⊥ ∩ b⊥ ) = (a ∪ b)⊥
42	41	ax-r1 35	. . . . 5 (a ∪ b)⊥ = (a⊥ ∩ b⊥ )
43		lear 161	. . . . . 6 ((a ∪ b) ∩ a) ≤ a
44		leo 158	. . . . . . 7 a ≤ (a ∪ b)
45		leid 148	. . . . . . 7 a ≤ a
46	44, 45	ler2an 173	. . . . . 6 a ≤ ((a ∪ b) ∩ a)
47	43, 46	lebi 145	. . . . 5 ((a ∪ b) ∩ a) = a
48	42, 47	2or 72	. . . 4 ((a ∪ b)⊥ ∪ ((a ∪ b) ∩ a)) = ((a⊥ ∩ b⊥ ) ∪ a)
49	40, 48	ax-r2 36	. . 3 ((a ∪ b) →1 a) = ((a⊥ ∩ b⊥ ) ∪ a)
50	39, 49	2an 79	. 2 ((a →1 (a ∩ b)) ∩ ((a ∪ b) →1 a)) = ((a⊥ ∪ (a ∩ b)) ∩ ((a⊥ ∩ b⊥ ) ∪ a))
51	9, 32, 50	3tr1 63	1 (a ≡ b) = ((a →1 (a ∩ b)) ∩ ((a ∪ b) →1 a))

Syntax hints:   = wb 1  ^⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7  0wf 9   →₁ wi1 12

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-i1 44  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by:  mlaconj  845