Theorem ni32 502

Description: Equivalence for Kalmbach implication.

Assertion

Ref	Expression
ni32	(a →3 b)⊥ = ((a ∪ b) ∩ ((a ∩ b⊥ ) ∪ (a⊥ ∩ (a ∪ b⊥ ))))

Proof of Theorem ni32

Step	Hyp	Ref	Expression
1		df2i3 498	. . 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	. . . . . . 7 (a ∪ b) = (a⊥ ∩ b⊥ )⊥
4		oran 87	. . . . . . . 8 ((a ∩ b⊥ ) ∪ (a⊥ ∩ (a ∪ b⊥ ))) = ((a ∩ b⊥ )⊥ ∩ (a⊥ ∩ (a ∪ b⊥ ))⊥ )⊥
5		anor1 88	. . . . . . . . . . . . 13 (a ∩ b⊥ ) = (a⊥ ∪ b)⊥
6	5	con2 67	. . . . . . . . . . . 12 (a ∩ b⊥ )⊥ = (a⊥ ∪ b)
7	6	ax-r1 35	. . . . . . . . . . 11 (a⊥ ∪ b) = (a ∩ b⊥ )⊥
8		oran 87	. . . . . . . . . . . 12 (a ∪ (a⊥ ∩ b)) = (a⊥ ∩ (a⊥ ∩ b)⊥ )⊥
9		anor2 89	. . . . . . . . . . . . . . 15 (a⊥ ∩ b) = (a ∪ b⊥ )⊥
10	9	con2 67	. . . . . . . . . . . . . 14 (a⊥ ∩ b)⊥ = (a ∪ b⊥ )
11	10	lan 77	. . . . . . . . . . . . 13 (a⊥ ∩ (a⊥ ∩ b)⊥ ) = (a⊥ ∩ (a ∪ b⊥ ))
12	11	ax-r4 37	. . . . . . . . . . . 12 (a⊥ ∩ (a⊥ ∩ b)⊥ )⊥ = (a⊥ ∩ (a ∪ b⊥ ))⊥
13	8, 12	ax-r2 36	. . . . . . . . . . 11 (a ∪ (a⊥ ∩ b)) = (a⊥ ∩ (a ∪ b⊥ ))⊥
14	7, 13	2an 79	. . . . . . . . . 10 ((a⊥ ∪ b) ∩ (a ∪ (a⊥ ∩ b))) = ((a ∩ b⊥ )⊥ ∩ (a⊥ ∩ (a ∪ b⊥ ))⊥ )
15	14	ax-r1 35	. . . . . . . . 9 ((a ∩ b⊥ )⊥ ∩ (a⊥ ∩ (a ∪ b⊥ ))⊥ ) = ((a⊥ ∪ b) ∩ (a ∪ (a⊥ ∩ b)))
16	15	ax-r4 37	. . . . . . . 8 ((a ∩ b⊥ )⊥ ∩ (a⊥ ∩ (a ∪ b⊥ ))⊥ )⊥ = ((a⊥ ∪ b) ∩ (a ∪ (a⊥ ∩ b)))⊥
17	4, 16	ax-r2 36	. . . . . . 7 ((a ∩ b⊥ ) ∪ (a⊥ ∩ (a ∪ b⊥ ))) = ((a⊥ ∪ b) ∩ (a ∪ (a⊥ ∩ b)))⊥
18	3, 17	2an 79	. . . . . 6 ((a ∪ b) ∩ ((a ∩ b⊥ ) ∪ (a⊥ ∩ (a ∪ b⊥ )))) = ((a⊥ ∩ b⊥ )⊥ ∩ ((a⊥ ∪ b) ∩ (a ∪ (a⊥ ∩ b)))⊥ )
19	18	ax-r1 35	. . . . 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-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-i3 46  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by:  oi3ai3  503  i3con  551