Theorem i3th7 549

Description: Theorem for Kalmbach implication. (Contributed by NM, 19-Nov-1997.)

Assertion

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

Proof of Theorem i3th7

Step	Hyp	Ref	Expression
1		leor 159	. . 3 a ≤ ((a →3 b)⊥ ∪ a)
2		lem4 511	. . . . 5 ((a →3 b) →3 ((a →3 b) →3 a)) = ((a →3 b)⊥ ∪ a)
3	2	ax-r1 35	. . . 4 ((a →3 b)⊥ ∪ a) = ((a →3 b) →3 ((a →3 b) →3 a))
4		i3abs3 524	. . . 4 ((a →3 b) →3 ((a →3 b) →3 a)) = ((a →3 b) →3 a)
5	3, 4	ax-r2 36	. . 3 ((a →3 b)⊥ ∪ a) = ((a →3 b) →3 a)
6	1, 5	lbtr 139	. 2 a ≤ ((a →3 b) →3 a)
7	6	lei3 246	1 (a →3 ((a →3 b) →3 a)) = 1

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6  1wt 8   →₃ wi3 14

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-r3 439

This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i3 46  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by: (None)