Theorem ni31 250

Description: Equivalence for Kalmbach implication.

Assertion

Ref	Expression
ni31	(a →3 b)⊥ = (((a ∪ b⊥ ) ∩ (a ∪ b)) ∩ (a⊥ ∪ (a ∩ b⊥ )))

Proof of Theorem ni31

Step	Hyp	Ref	Expression
1		df-i3 46	. . 3 (a →3 b) = (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b)))
2		oran 87	. . . 4 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))) = (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))⊥ ∩ (a ∩ (a⊥ ∪ b))⊥ )⊥
3		oran 87	. . . . . . . 8 ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) = ((a⊥ ∩ b)⊥ ∩ (a⊥ ∩ b⊥ )⊥ )⊥
4		anor2 89	. . . . . . . . . . 11 (a⊥ ∩ b) = (a ∪ b⊥ )⊥
5	4	con2 67	. . . . . . . . . 10 (a⊥ ∩ b)⊥ = (a ∪ b⊥ )
6		oran 87	. . . . . . . . . . 11 (a ∪ b) = (a⊥ ∩ b⊥ )⊥
7	6	ax-r1 35	. . . . . . . . . 10 (a⊥ ∩ b⊥ )⊥ = (a ∪ b)
8	5, 7	2an 79	. . . . . . . . 9 ((a⊥ ∩ b)⊥ ∩ (a⊥ ∩ b⊥ )⊥ ) = ((a ∪ b⊥ ) ∩ (a ∪ b))
9	8	ax-r4 37	. . . . . . . 8 ((a⊥ ∩ b)⊥ ∩ (a⊥ ∩ b⊥ )⊥ )⊥ = ((a ∪ b⊥ ) ∩ (a ∪ b))⊥
10	3, 9	ax-r2 36	. . . . . . 7 ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) = ((a ∪ b⊥ ) ∩ (a ∪ b))⊥
11	10	con2 67	. . . . . 6 ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))⊥ = ((a ∪ b⊥ ) ∩ (a ∪ b))
12		df-a 40	. . . . . . . 8 (a ∩ (a⊥ ∪ b)) = (a⊥ ∪ (a⊥ ∪ b)⊥ )⊥
13		anor1 88	. . . . . . . . . . 11 (a ∩ b⊥ ) = (a⊥ ∪ b)⊥
14	13	ax-r1 35	. . . . . . . . . 10 (a⊥ ∪ b)⊥ = (a ∩ b⊥ )
15	14	lor 70	. . . . . . . . 9 (a⊥ ∪ (a⊥ ∪ b)⊥ ) = (a⊥ ∪ (a ∩ b⊥ ))
16	15	ax-r4 37	. . . . . . . 8 (a⊥ ∪ (a⊥ ∪ b)⊥ )⊥ = (a⊥ ∪ (a ∩ b⊥ ))⊥
17	12, 16	ax-r2 36	. . . . . . 7 (a ∩ (a⊥ ∪ b)) = (a⊥ ∪ (a ∩ b⊥ ))⊥
18	17	con2 67	. . . . . 6 (a ∩ (a⊥ ∪ b))⊥ = (a⊥ ∪ (a ∩ b⊥ ))
19	11, 18	2an 79	. . . . 5 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))⊥ ∩ (a ∩ (a⊥ ∪ b))⊥ ) = (((a ∪ b⊥ ) ∩ (a ∪ b)) ∩ (a⊥ ∪ (a ∩ b⊥ )))
20	19	ax-r4 37	. . . 4 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))⊥ ∩ (a ∩ (a⊥ ∪ b))⊥ )⊥ = (((a ∪ b⊥ ) ∩ (a ∪ b)) ∩ (a⊥ ∪ (a ∩ b⊥ )))⊥
21	2, 20	ax-r2 36	. . 3 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))) = (((a ∪ b⊥ ) ∩ (a ∪ b)) ∩ (a⊥ ∪ (a ∩ b⊥ )))⊥
22	1, 21	ax-r2 36	. 2 (a →3 b) = (((a ∪ b⊥ ) ∩ (a ∪ b)) ∩ (a⊥ ∪ (a ∩ b⊥ )))⊥
23	22	con2 67	1 (a →3 b)⊥ = (((a ∪ b⊥ ) ∩ (a ∪ b)) ∩ (a⊥ ∪ (a ∩ b⊥ )))

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₃ wi3 14

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38

This theorem depends on definitions:  df-a 40  df-i3 46

This theorem is referenced by:  ud3lem0c  279