Theorem u12lembi 708

Description: Sasaki/Dishkant implication and biconditional.

Assertion

Ref	Expression
u12lembi	((a →1 b) ∩ (b →2 a)) = (a ≡ b)

Proof of Theorem u12lembi

Step	Hyp	Ref	Expression
1		u1lemc1 662	. . . . 5 a C (a →1 b)
2	1	comcom 435	. . . 4 (a →1 b) C a
3		lear 153	. . . . . . 7 (b⊥ ∩ a⊥ ) ≤ a⊥
4		leo 150	. . . . . . . 8 a⊥ ≤ (a⊥ ∪ (a ∩ b))
5		df-i1 43	. . . . . . . . 9 (a →1 b) = (a⊥ ∪ (a ∩ b))
6	5	ax-r1 34	. . . . . . . 8 (a⊥ ∪ (a ∩ b)) = (a →1 b)
7	4, 6	lbtr 131	. . . . . . 7 a⊥ ≤ (a →1 b)
8	3, 7	letr 129	. . . . . 6 (b⊥ ∩ a⊥ ) ≤ (a →1 b)
9	8	lecom 172	. . . . 5 (b⊥ ∩ a⊥ ) C (a →1 b)
10	9	comcom 435	. . . 4 (a →1 b) C (b⊥ ∩ a⊥ )
11	2, 10	fh1 451	. . 3 ((a →1 b) ∩ (a ∪ (b⊥ ∩ a⊥ ))) = (((a →1 b) ∩ a) ∪ ((a →1 b) ∩ (b⊥ ∩ a⊥ )))
12		u1lemaa 582	. . . 4 ((a →1 b) ∩ a) = (a ∩ b)
13		an12 74	. . . . 5 ((a →1 b) ∩ (b⊥ ∩ a⊥ )) = (b⊥ ∩ ((a →1 b) ∩ a⊥ ))
14		u1lemana 587	. . . . . 6 ((a →1 b) ∩ a⊥ ) = a⊥
15	14	lan 70	. . . . 5 (b⊥ ∩ ((a →1 b) ∩ a⊥ )) = (b⊥ ∩ a⊥ )
16		ancom 68	. . . . 5 (b⊥ ∩ a⊥ ) = (a⊥ ∩ b⊥ )
17	13, 15, 16	3tr 62	. . . 4 ((a →1 b) ∩ (b⊥ ∩ a⊥ )) = (a⊥ ∩ b⊥ )
18	12, 17	2or 67	. . 3 (((a →1 b) ∩ a) ∪ ((a →1 b) ∩ (b⊥ ∩ a⊥ ))) = ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))
19	11, 18	ax-r2 35	. 2 ((a →1 b) ∩ (a ∪ (b⊥ ∩ a⊥ ))) = ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))
20		df-i2 44	. . 3 (b →2 a) = (a ∪ (b⊥ ∩ a⊥ ))
21	20	lan 70	. 2 ((a →1 b) ∩ (b →2 a)) = ((a →1 b) ∩ (a ∪ (b⊥ ∩ a⊥ )))
22		dfb 86	. 2 (a ≡ b) = ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))
23	19, 21, 22	3tr1 60	1 ((a →1 b) ∩ (b →2 a)) = (a ≡ b)

Syntax hints:   = wb 1  ^⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7   →₁ wi1 13   →₂ wi2 14

This theorem is referenced by:  bi3 821  bi4 822

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-i2 44  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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