Theorem u3lem9 784

Description: Lemma for unified implication study. (Contributed by NM, 24-Dec-1997.)

Assertion

Ref	Expression
u3lem9	(a →3 (a →3 (a⊥ →3 b))) = (a →3 (a⊥ →3 b))

Proof of Theorem u3lem9

Step	Hyp	Ref	Expression
1		comi31 508	. . 3 a C (a →3 (a⊥ →3 b))
2	1	u3lemc4 703	. 2 (a →3 (a →3 (a⊥ →3 b))) = (a⊥ ∪ (a →3 (a⊥ →3 b)))
3		u3lem7 774	. . . 4 (a →3 (a⊥ →3 b)) = (a⊥ ∪ ((a ∩ b) ∪ (a ∩ b⊥ )))
4	3	lor 70	. . 3 (a⊥ ∪ (a →3 (a⊥ →3 b))) = (a⊥ ∪ (a⊥ ∪ ((a ∩ b) ∪ (a ∩ b⊥ ))))
5		ax-a3 32	. . . . 5 ((a⊥ ∪ a⊥ ) ∪ ((a ∩ b) ∪ (a ∩ b⊥ ))) = (a⊥ ∪ (a⊥ ∪ ((a ∩ b) ∪ (a ∩ b⊥ ))))
6	5	ax-r1 35	. . . 4 (a⊥ ∪ (a⊥ ∪ ((a ∩ b) ∪ (a ∩ b⊥ )))) = ((a⊥ ∪ a⊥ ) ∪ ((a ∩ b) ∪ (a ∩ b⊥ )))
7		oridm 110	. . . . . 6 (a⊥ ∪ a⊥ ) = a⊥
8	7	ax-r5 38	. . . . 5 ((a⊥ ∪ a⊥ ) ∪ ((a ∩ b) ∪ (a ∩ b⊥ ))) = (a⊥ ∪ ((a ∩ b) ∪ (a ∩ b⊥ )))
9	3	ax-r1 35	. . . . 5 (a⊥ ∪ ((a ∩ b) ∪ (a ∩ b⊥ ))) = (a →3 (a⊥ →3 b))
10	8, 9	ax-r2 36	. . . 4 ((a⊥ ∪ a⊥ ) ∪ ((a ∩ b) ∪ (a ∩ b⊥ ))) = (a →3 (a⊥ →3 b))
11	6, 10	ax-r2 36	. . 3 (a⊥ ∪ (a⊥ ∪ ((a ∩ b) ∪ (a ∩ b⊥ )))) = (a →3 (a⊥ →3 b))
12	4, 11	ax-r2 36	. 2 (a⊥ ∪ (a →3 (a⊥ →3 b))) = (a →3 (a⊥ →3 b))
13	2, 12	ax-r2 36	1 (a →3 (a →3 (a⊥ →3 b))) = (a →3 (a⊥ →3 b))

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