Theorem i3th5 529

Description: Theorem for Kalmbach implication.

Assertion

Ref	Expression
i3th5	((a ->3 b) ->3 (a ->3 (a ->3 b))) = 1

Proof of Theorem i3th5

Step	Hyp	Ref	Expression
1		ax-a2 30	. . . . . 6 ((a_|_ ^ b) v (a_|_ ^ b_|_)) = ((a_|_ ^ b_|_) v (a_|_ ^ b))
2		lea 152	. . . . . . 7 (a_|_ ^ b_|_) =< a_|_
3		lear 153	. . . . . . 7 (a_|_ ^ b) =< b
4	2, 3	le2or 160	. . . . . 6 ((a_|_ ^ b_|_) v (a_|_ ^ b)) =< (a_|_ v b)
5	1, 4	bltr 130	. . . . 5 ((a_|_ ^ b) v (a_|_ ^ b_|_)) =< (a_|_ v b)
6		lear 153	. . . . 5 (a ^ (a_|_ v b)) =< (a_|_ v b)
7	5, 6	le2or 160	. . . 4 (((a_|_ ^ b) v (a_|_ ^ b_|_)) v (a ^ (a_|_ v b))) =< ((a_|_ v b) v (a_|_ v b))
8		oridm 102	. . . 4 ((a_|_ v b) v (a_|_ v b)) = (a_|_ v b)
9	7, 8	lbtr 131	. . 3 (((a_|_ ^ b) v (a_|_ ^ b_|_)) v (a ^ (a_|_ v b))) =< (a_|_ v b)
10		df-i3 45	. . 3 (a ->3 b) = (((a_|_ ^ b) v (a_|_ ^ b_|_)) v (a ^ (a_|_ v b)))
11		lem4 493	. . 3 (a ->3 (a ->3 b)) = (a_|_ v b)
12	9, 10, 11	le3tr1 132	. 2 (a ->3 b) =< (a ->3 (a ->3 b))
13	12	lei3 238	1 ((a ->3 b) ->3 (a ->3 (a ->3 b))) = 1

Syntax hints:   = wb 1  _|_wn 4   v wo 6   ^ wa 7  1wt 9   ->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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