Theorem ni32 484

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 480	. . 3 (a →3 b) = ((a⊥ ∩ b⊥ ) ∪ ((a⊥ ∪ b) ∩ (a ∪ (a⊥ ∩ b))))
2		oran 79	. . . 4 ((a⊥ ∩ b⊥ ) ∪ ((a⊥ ∪ b) ∩ (a ∪ (a⊥ ∩ b)))) = ((a⊥ ∩ b⊥ )⊥ ∩ ((a⊥ ∪ b) ∩ (a ∪ (a⊥ ∩ b)))⊥ )⊥
3		oran 79	. . . . . . 7 (a ∪ b) = (a⊥ ∩ b⊥ )⊥
4		oran 79	. . . . . . . 8 ((a ∩ b⊥ ) ∪ (a⊥ ∩ (a ∪ b⊥ ))) = ((a ∩ b⊥ )⊥ ∩ (a⊥ ∩ (a ∪ b⊥ ))⊥ )⊥
5		anor1 80	. . . . . . . . . . . . 13 (a ∩ b⊥ ) = (a⊥ ∪ b)⊥
6	5	con2 64	. . . . . . . . . . . 12 (a ∩ b⊥ )⊥ = (a⊥ ∪ b)
7	6	ax-r1 34	. . . . . . . . . . 11 (a⊥ ∪ b) = (a ∩ b⊥ )⊥
8		oran 79	. . . . . . . . . . . 12 (a ∪ (a⊥ ∩ b)) = (a⊥ ∩ (a⊥ ∩ b)⊥ )⊥
9		anor2 81	. . . . . . . . . . . . . . 15 (a⊥ ∩ b) = (a ∪ b⊥ )⊥
10	9	con2 64	. . . . . . . . . . . . . 14 (a⊥ ∩ b)⊥ = (a ∪ b⊥ )
11	10	lan 70	. . . . . . . . . . . . 13 (a⊥ ∩ (a⊥ ∩ b)⊥ ) = (a⊥ ∩ (a ∪ b⊥ ))
12	11	ax-r4 36	. . . . . . . . . . . 12 (a⊥ ∩ (a⊥ ∩ b)⊥ )⊥ = (a⊥ ∩ (a ∪ b⊥ ))⊥
13	8, 12	ax-r2 35	. . . . . . . . . . 11 (a ∪ (a⊥ ∩ b)) = (a⊥ ∩ (a ∪ b⊥ ))⊥
14	7, 13	2an 72	. . . . . . . . . 10 ((a⊥ ∪ b) ∩ (a ∪ (a⊥ ∩ b))) = ((a ∩ b⊥ )⊥ ∩ (a⊥ ∩ (a ∪ b⊥ ))⊥ )
15	14	ax-r1 34	. . . . . . . . 9 ((a ∩ b⊥ )⊥ ∩ (a⊥ ∩ (a ∪ b⊥ ))⊥ ) = ((a⊥ ∪ b) ∩ (a ∪ (a⊥ ∩ b)))
16	15	ax-r4 36	. . . . . . . 8 ((a ∩ b⊥ )⊥ ∩ (a⊥ ∩ (a ∪ b⊥ ))⊥ )⊥ = ((a⊥ ∪ b) ∩ (a ∪ (a⊥ ∩ b)))⊥
17	4, 16	ax-r2 35	. . . . . . 7 ((a ∩ b⊥ ) ∪ (a⊥ ∩ (a ∪ b⊥ ))) = ((a⊥ ∪ b) ∩ (a ∪ (a⊥ ∩ b)))⊥
18	3, 17	2an 72	. . . . . 6 ((a ∪ b) ∩ ((a ∩ b⊥ ) ∪ (a⊥ ∩ (a ∪ b⊥ )))) = ((a⊥ ∩ b⊥ )⊥ ∩ ((a⊥ ∪ b) ∩ (a ∪ (a⊥ ∩ b)))⊥ )
19	18	ax-r1 34	. . . . 5 ((a⊥ ∩ b⊥ )⊥ ∩ ((a⊥ ∪ b) ∩ (a ∪ (a⊥ ∩ b)))⊥ ) = ((a ∪ b) ∩ ((a ∩ b⊥ ) ∪ (a⊥ ∩ (a ∪ b⊥ ))))
20	19	ax-r4 36	. . . 4 ((a⊥ ∩ b⊥ )⊥ ∩ ((a⊥ ∪ b) ∩ (a ∪ (a⊥ ∩ b)))⊥ )⊥ = ((a ∪ b) ∩ ((a ∩ b⊥ ) ∪ (a⊥ ∩ (a ∪ b⊥ ))))⊥
21	2, 20	ax-r2 35	. . 3 ((a⊥ ∩ b⊥ ) ∪ ((a⊥ ∪ b) ∩ (a ∪ (a⊥ ∩ b)))) = ((a ∪ b) ∩ ((a ∩ b⊥ ) ∪ (a⊥ ∩ (a ∪ b⊥ ))))⊥
22	1, 21	ax-r2 35	. 2 (a →3 b) = ((a ∪ b) ∩ ((a ∩ b⊥ ) ∪ (a⊥ ∩ (a ∪ b⊥ ))))⊥
23	22	con2 64	1 (a →3 b)⊥ = ((a ∪ b) ∩ ((a ∩ b⊥ ) ∪ (a⊥ ∩ (a ∪ b⊥ ))))

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₃ wi3 15

This theorem is referenced by:  oi3ai3 485  i3con 533

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-i3 45  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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