Theorem u12lembi 726

Description: Sasaki/Dishkant implication and biconditional. Equation 4.14 of [MegPav2000] p. 23. The variable i in the paper is set to 1, and j is set to 2. (Contributed by NM, 2-Mar-2000.)

Assertion

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

Proof of Theorem u12lembi

Step	Hyp	Ref	Expression
1		u1lemc1 680	. . . . 5 a C (a →1 b)
2	1	comcom 453	. . . 4 (a →1 b) C a
3		lear 161	. . . . . . 7 (b⊥ ∩ a⊥ ) ≤ a⊥
4		leo 158	. . . . . . . 8 a⊥ ≤ (a⊥ ∪ (a ∩ b))
5		df-i1 44	. . . . . . . . 9 (a →1 b) = (a⊥ ∪ (a ∩ b))
6	5	ax-r1 35	. . . . . . . 8 (a⊥ ∪ (a ∩ b)) = (a →1 b)
7	4, 6	lbtr 139	. . . . . . 7 a⊥ ≤ (a →1 b)
8	3, 7	letr 137	. . . . . 6 (b⊥ ∩ a⊥ ) ≤ (a →1 b)
9	8	lecom 180	. . . . 5 (b⊥ ∩ a⊥ ) C (a →1 b)
10	9	comcom 453	. . . 4 (a →1 b) C (b⊥ ∩ a⊥ )
11	2, 10	fh1 469	. . 3 ((a →1 b) ∩ (a ∪ (b⊥ ∩ a⊥ ))) = (((a →1 b) ∩ a) ∪ ((a →1 b) ∩ (b⊥ ∩ a⊥ )))
12		u1lemaa 600	. . . 4 ((a →1 b) ∩ a) = (a ∩ b)
13		an12 81	. . . . 5 ((a →1 b) ∩ (b⊥ ∩ a⊥ )) = (b⊥ ∩ ((a →1 b) ∩ a⊥ ))
14		u1lemana 605	. . . . . 6 ((a →1 b) ∩ a⊥ ) = a⊥
15	14	lan 77	. . . . 5 (b⊥ ∩ ((a →1 b) ∩ a⊥ )) = (b⊥ ∩ a⊥ )
16		ancom 74	. . . . 5 (b⊥ ∩ a⊥ ) = (a⊥ ∩ b⊥ )
17	13, 15, 16	3tr 65	. . . 4 ((a →1 b) ∩ (b⊥ ∩ a⊥ )) = (a⊥ ∩ b⊥ )
18	12, 17	2or 72	. . 3 (((a →1 b) ∩ a) ∪ ((a →1 b) ∩ (b⊥ ∩ a⊥ ))) = ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))
19	11, 18	ax-r2 36	. 2 ((a →1 b) ∩ (a ∪ (b⊥ ∩ a⊥ ))) = ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))
20		df-i2 45	. . 3 (b →2 a) = (a ∪ (b⊥ ∩ a⊥ ))
21	20	lan 77	. 2 ((a →1 b) ∩ (b →2 a)) = ((a →1 b) ∩ (a ∪ (b⊥ ∩ a⊥ )))
22		dfb 94	. 2 (a ≡ b) = ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))
23	19, 21, 22	3tr1 63	1 ((a →1 b) ∩ (b →2 a)) = (a ≡ b)

Syntax hints:   = wb 1  ^⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7   →₁ wi1 12   →₂ wi2 13

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-i2 45  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by:  bi3  839  bi4  840