Theorem i3n2 483

Description: Equivalence for Kalmbach implication.

Assertion

Ref	Expression
i3n2	(a_|_ ->3 b_|_) = ((a ^ b) v ((a v b_|_) ^ (a_|_ v (a ^ b_|_))))

Proof of Theorem i3n2

Step	Hyp	Ref	Expression
1		df2i3 480	. 2 (a_|_ ->3 b_|_) = ((a_|__|_ ^ b_|__|_) v ((a_|__|_ v b_|_) ^ (a_|_ v (a_|__|_ ^ b_|_))))
2		ax-a1 29	. . . . 5 a = a_|__|_
3		ax-a1 29	. . . . 5 b = b_|__|_
4	2, 3	2an 72	. . . 4 (a ^ b) = (a_|__|_ ^ b_|__|_)
5	2	ax-r5 37	. . . . 5 (a v b_|_) = (a_|__|_ v b_|_)
6	2	ran 71	. . . . . 6 (a ^ b_|_) = (a_|__|_ ^ b_|_)
7	6	lor 66	. . . . 5 (a_|_ v (a ^ b_|_)) = (a_|_ v (a_|__|_ ^ b_|_))
8	5, 7	2an 72	. . . 4 ((a v b_|_) ^ (a_|_ v (a ^ b_|_))) = ((a_|__|_ v b_|_) ^ (a_|_ v (a_|__|_ ^ b_|_)))
9	4, 8	2or 67	. . 3 ((a ^ b) v ((a v b_|_) ^ (a_|_ v (a ^ b_|_)))) = ((a_|__|_ ^ b_|__|_) v ((a_|__|_ v b_|_) ^ (a_|_ v (a_|__|_ ^ b_|_))))
10	9	ax-r1 34	. 2 ((a_|__|_ ^ b_|__|_) v ((a_|__|_ v b_|_) ^ (a_|_ v (a_|__|_ ^ b_|_)))) = ((a ^ b) v ((a v b_|_) ^ (a_|_ v (a ^ b_|_))))
11	1, 10	ax-r2 35	1 (a_|_ ->3 b_|_) = ((a ^ b) v ((a v b_|_) ^ (a_|_ v (a ^ b_|_))))

Syntax hints:   = wb 1  _|_wn 4   v wo 6   ^ wa 7   ->3 wi3 15

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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