Theorem i3th7 531

Description: Theorem for Kalmbach implication.

Assertion

Ref	Expression
i3th7	(a →3 ((a →3 b) →3 a)) = 1

Proof of Theorem i3th7

Step	Hyp	Ref	Expression
1		leor 151	. . 3 a ≤ ((a →3 b)⊥ ∪ a)
2		lem4 493	. . . . 5 ((a →3 b) →3 ((a →3 b) →3 a)) = ((a →3 b)⊥ ∪ a)
3	2	ax-r1 34	. . . 4 ((a →3 b)⊥ ∪ a) = ((a →3 b) →3 ((a →3 b) →3 a))
4		i3abs3 506	. . . 4 ((a →3 b) →3 ((a →3 b) →3 a)) = ((a →3 b) →3 a)
5	3, 4	ax-r2 35	. . 3 ((a →3 b)⊥ ∪ a) = ((a →3 b) →3 a)
6	1, 5	lbtr 131	. 2 a ≤ ((a →3 b) →3 a)
7	6	lei3 238	1 (a →3 ((a →3 b) →3 a)) = 1

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6  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

metamath.org