Theorem i3th5 547

Description: Theorem for Kalmbach implication. (Contributed by NM, 16-Nov-1997.)

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 31	. . . . . 6 ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) = ((a⊥ ∩ b⊥ ) ∪ (a⊥ ∩ b))
2		lea 160	. . . . . . 7 (a⊥ ∩ b⊥ ) ≤ a⊥
3		lear 161	. . . . . . 7 (a⊥ ∩ b) ≤ b
4	2, 3	le2or 168	. . . . . 6 ((a⊥ ∩ b⊥ ) ∪ (a⊥ ∩ b)) ≤ (a⊥ ∪ b)
5	1, 4	bltr 138	. . . . 5 ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ≤ (a⊥ ∪ b)
6		lear 161	. . . . 5 (a ∩ (a⊥ ∪ b)) ≤ (a⊥ ∪ b)
7	5, 6	le2or 168	. . . 4 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))) ≤ ((a⊥ ∪ b) ∪ (a⊥ ∪ b))
8		oridm 110	. . . 4 ((a⊥ ∪ b) ∪ (a⊥ ∪ b)) = (a⊥ ∪ b)
9	7, 8	lbtr 139	. . 3 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))) ≤ (a⊥ ∪ b)
10		df-i3 46	. . 3 (a →3 b) = (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b)))
11		lem4 511	. . 3 (a →3 (a →3 b)) = (a⊥ ∪ b)
12	9, 10, 11	le3tr1 140	. 2 (a →3 b) ≤ (a →3 (a →3 b))
13	12	lei3 246	1 ((a →3 b) →3 (a →3 (a →3 b))) = 1

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  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)