Theorem i2i1i1 782

Description: Equivalence to →₂ .

Assertion

Ref	Expression
i2i1i1	(a →2 b) = ((a →1 (a ∪ b)) ∩ ((a ∪ b) →1 b))

Proof of Theorem i2i1i1

Step	Hyp	Ref	Expression
1		an1r 99	. . 3 (1 ∩ (b ∪ (a⊥ ∩ b⊥ ))) = (b ∪ (a⊥ ∩ b⊥ ))
2	1	ax-r1 34	. 2 (b ∪ (a⊥ ∩ b⊥ )) = (1 ∩ (b ∪ (a⊥ ∩ b⊥ )))
3		df-i2 44	. 2 (a →2 b) = (b ∪ (a⊥ ∩ b⊥ ))
4		a5c 113	. . . . . 6 (a ∩ (a ∪ b)) = a
5	4	lor 66	. . . . 5 (a⊥ ∪ (a ∩ (a ∪ b))) = (a⊥ ∪ a)
6		ax-a2 30	. . . . 5 (a⊥ ∪ a) = (a ∪ a⊥ )
7	5, 6	ax-r2 35	. . . 4 (a⊥ ∪ (a ∩ (a ∪ b))) = (a ∪ a⊥ )
8		df-i1 43	. . . 4 (a →1 (a ∪ b)) = (a⊥ ∪ (a ∩ (a ∪ b)))
9		df-t 40	. . . 4 1 = (a ∪ a⊥ )
10	7, 8, 9	3tr1 60	. . 3 (a →1 (a ∪ b)) = 1
11		df-i1 43	. . . 4 ((a ∪ b) →1 b) = ((a ∪ b)⊥ ∪ ((a ∪ b) ∩ b))
12		anor3 82	. . . . . 6 (a⊥ ∩ b⊥ ) = (a ∪ b)⊥
13		leor 151	. . . . . . . 8 b ≤ (a ∪ b)
14		leid 140	. . . . . . . 8 b ≤ b
15	13, 14	ler2an 165	. . . . . . 7 b ≤ ((a ∪ b) ∩ b)
16		lear 153	. . . . . . 7 ((a ∪ b) ∩ b) ≤ b
17	15, 16	lebi 137	. . . . . 6 b = ((a ∪ b) ∩ b)
18	12, 17	2or 67	. . . . 5 ((a⊥ ∩ b⊥ ) ∪ b) = ((a ∪ b)⊥ ∪ ((a ∪ b) ∩ b))
19	18	ax-r1 34	. . . 4 ((a ∪ b)⊥ ∪ ((a ∪ b) ∩ b)) = ((a⊥ ∩ b⊥ ) ∪ b)
20		ax-a2 30	. . . 4 ((a⊥ ∩ b⊥ ) ∪ b) = (b ∪ (a⊥ ∩ b⊥ ))
21	11, 19, 20	3tr 62	. . 3 ((a ∪ b) →1 b) = (b ∪ (a⊥ ∩ b⊥ ))
22	10, 21	2an 72	. 2 ((a →1 (a ∪ b)) ∩ ((a ∪ b) →1 b)) = (1 ∩ (b ∪ (a⊥ ∩ b⊥ )))
23	2, 3, 22	3tr1 60	1 (a →2 b) = ((a →1 (a ∪ b)) ∩ ((a ∪ b) →1 b))

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 9   →₁ wi1 13   →₂ wi2 14

This theorem is referenced by:  mlaconj 827

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37

This theorem depends on definitions:  df-a 39  df-t 40  df-f 41  df-i1 43  df-i2 44  df-le1 122  df-le2 123

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