Theorem i3n1 241

Description: Equivalence for Kalmbach implication.

Assertion

Ref	Expression
i3n1	(a_|_ ->3 b_|_) = (((a ^ b_|_) v (a ^ b)) v (a_|_ ^ (a v b_|_)))

Proof of Theorem i3n1

Step	Hyp	Ref	Expression
1		df-i3 45	. 2 (a_|_ ->3 b_|_) = (((a_|__|_ ^ b_|_) v (a_|__|_ ^ b_|__|_)) v (a_|_ ^ (a_|__|_ v b_|_)))
2		ax-a1 29	. . . . . 6 a = a_|__|_
3	2	ran 71	. . . . 5 (a ^ b_|_) = (a_|__|_ ^ b_|_)
4		ax-a1 29	. . . . . 6 b = b_|__|_
5	2, 4	2an 72	. . . . 5 (a ^ b) = (a_|__|_ ^ b_|__|_)
6	3, 5	2or 67	. . . 4 ((a ^ b_|_) v (a ^ b)) = ((a_|__|_ ^ b_|_) v (a_|__|_ ^ b_|__|_))
7	2	ax-r5 37	. . . . 5 (a v b_|_) = (a_|__|_ v b_|_)
8	7	lan 70	. . . 4 (a_|_ ^ (a v b_|_)) = (a_|_ ^ (a_|__|_ v b_|_))
9	6, 8	2or 67	. . 3 (((a ^ b_|_) v (a ^ b)) v (a_|_ ^ (a v b_|_))) = (((a_|__|_ ^ b_|_) v (a_|__|_ ^ b_|__|_)) v (a_|_ ^ (a_|__|_ v b_|_)))
10	9	ax-r1 34	. 2 (((a_|__|_ ^ b_|_) v (a_|__|_ ^ b_|__|_)) v (a_|_ ^ (a_|__|_ v b_|_))) = (((a ^ b_|_) v (a ^ b)) v (a_|_ ^ (a v b_|_)))
11	1, 10	ax-r2 35	1 (a_|_ ->3 b_|_) = (((a ^ b_|_) v (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  i3orlem7 540  i3orlem8 541

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37

This theorem depends on definitions:  df-a 39  df-i3 45

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