Theorem ni32 484

Description: Equivalence for Kalmbach implication.

Assertion

Ref	Expression
ni32	(a ->3 b)_|_ = ((a v b) ^ ((a ^ b_|_) v (a_|_ ^ (a v b_|_))))

Proof of Theorem ni32

Step	Hyp	Ref	Expression
1		df2i3 480	. . 3 (a ->3 b) = ((a_|_ ^ b_|_) v ((a_|_ v b) ^ (a v (a_|_ ^ b))))
2		oran 79	. . . 4 ((a_|_ ^ b_|_) v ((a_|_ v b) ^ (a v (a_|_ ^ b)))) = ((a_|_ ^ b_|_)_|_ ^ ((a_|_ v b) ^ (a v (a_|_ ^ b)))_|_)_|_
3		oran 79	. . . . . . 7 (a v b) = (a_|_ ^ b_|_)_|_
4		oran 79	. . . . . . . 8 ((a ^ b_|_) v (a_|_ ^ (a v b_|_))) = ((a ^ b_|_)_|_ ^ (a_|_ ^ (a v b_|_))_|_)_|_
5		anor1 80	. . . . . . . . . . . . 13 (a ^ b_|_) = (a_|_ v b)_|_
6	5	con2 64	. . . . . . . . . . . 12 (a ^ b_|_)_|_ = (a_|_ v b)
7	6	ax-r1 34	. . . . . . . . . . 11 (a_|_ v b) = (a ^ b_|_)_|_
8		oran 79	. . . . . . . . . . . 12 (a v (a_|_ ^ b)) = (a_|_ ^ (a_|_ ^ b)_|_)_|_
9		anor2 81	. . . . . . . . . . . . . . 15 (a_|_ ^ b) = (a v b_|_)_|_
10	9	con2 64	. . . . . . . . . . . . . 14 (a_|_ ^ b)_|_ = (a v b_|_)
11	10	lan 70	. . . . . . . . . . . . 13 (a_|_ ^ (a_|_ ^ b)_|_) = (a_|_ ^ (a v b_|_))
12	11	ax-r4 36	. . . . . . . . . . . 12 (a_|_ ^ (a_|_ ^ b)_|_)_|_ = (a_|_ ^ (a v b_|_))_|_
13	8, 12	ax-r2 35	. . . . . . . . . . 11 (a v (a_|_ ^ b)) = (a_|_ ^ (a v b_|_))_|_
14	7, 13	2an 72	. . . . . . . . . 10 ((a_|_ v b) ^ (a v (a_|_ ^ b))) = ((a ^ b_|_)_|_ ^ (a_|_ ^ (a v b_|_))_|_)
15	14	ax-r1 34	. . . . . . . . 9 ((a ^ b_|_)_|_ ^ (a_|_ ^ (a v b_|_))_|_) = ((a_|_ v b) ^ (a v (a_|_ ^ b)))
16	15	ax-r4 36	. . . . . . . 8 ((a ^ b_|_)_|_ ^ (a_|_ ^ (a v b_|_))_|_)_|_ = ((a_|_ v b) ^ (a v (a_|_ ^ b)))_|_
17	4, 16	ax-r2 35	. . . . . . 7 ((a ^ b_|_) v (a_|_ ^ (a v b_|_))) = ((a_|_ v b) ^ (a v (a_|_ ^ b)))_|_
18	3, 17	2an 72	. . . . . 6 ((a v b) ^ ((a ^ b_|_) v (a_|_ ^ (a v b_|_)))) = ((a_|_ ^ b_|_)_|_ ^ ((a_|_ v b) ^ (a v (a_|_ ^ b)))_|_)
19	18	ax-r1 34	. . . . 5 ((a_|_ ^ b_|_)_|_ ^ ((a_|_ v b) ^ (a v (a_|_ ^ b)))_|_) = ((a v b) ^ ((a ^ b_|_) v (a_|_ ^ (a v b_|_))))
20	19	ax-r4 36	. . . 4 ((a_|_ ^ b_|_)_|_ ^ ((a_|_ v b) ^ (a v (a_|_ ^ b)))_|_)_|_ = ((a v b) ^ ((a ^ b_|_) v (a_|_ ^ (a v b_|_))))_|_
21	2, 20	ax-r2 35	. . 3 ((a_|_ ^ b_|_) v ((a_|_ v b) ^ (a v (a_|_ ^ b)))) = ((a v b) ^ ((a ^ b_|_) v (a_|_ ^ (a v b_|_))))_|_
22	1, 21	ax-r2 35	. 2 (a ->3 b) = ((a v b) ^ ((a ^ b_|_) v (a_|_ ^ (a v b_|_))))_|_
23	22	con2 64	1 (a ->3 b)_|_ = ((a v b) ^ ((a ^ b_|_) v (a_|_ ^ (a v b_|_))))

Syntax hints:   = wb 1  _|_wn 4   v 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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