Theorem i3abs1 522

Description: Antecedent absorption. (Contributed by NM, 16-Nov-1997.)

Assertion

Ref	Expression
i3abs1	(a →3 (a →3 (a →3 b))) = (a →3 (a →3 b))

Proof of Theorem i3abs1

Step	Hyp	Ref	Expression
1		orordi 112	. . . . 5 (a⊥ ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) = ((a⊥ ∪ (a⊥ ∩ b)) ∪ (a⊥ ∪ (a⊥ ∩ b⊥ )))
2		orabs 120	. . . . . . 7 (a⊥ ∪ (a⊥ ∩ b)) = a⊥
3		orabs 120	. . . . . . 7 (a⊥ ∪ (a⊥ ∩ b⊥ )) = a⊥
4	2, 3	2or 72	. . . . . 6 ((a⊥ ∪ (a⊥ ∩ b)) ∪ (a⊥ ∪ (a⊥ ∩ b⊥ ))) = (a⊥ ∪ a⊥ )
5		oridm 110	. . . . . 6 (a⊥ ∪ a⊥ ) = a⊥
6	4, 5	ax-r2 36	. . . . 5 ((a⊥ ∪ (a⊥ ∩ b)) ∪ (a⊥ ∪ (a⊥ ∩ b⊥ ))) = a⊥
7	1, 6	ax-r2 36	. . . 4 (a⊥ ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) = a⊥
8	7	ax-r5 38	. . 3 ((a⊥ ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) ∪ (a ∩ (a⊥ ∪ b))) = (a⊥ ∪ (a ∩ (a⊥ ∪ b)))
9		ax-a3 32	. . 3 ((a⊥ ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) ∪ (a ∩ (a⊥ ∪ b))) = (a⊥ ∪ (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))))
10		omln 446	. . 3 (a⊥ ∪ (a ∩ (a⊥ ∪ b))) = (a⊥ ∪ b)
11	8, 9, 10	3tr2 64	. 2 (a⊥ ∪ (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b)))) = (a⊥ ∪ b)
12		lem4 511	. . 3 (a →3 (a →3 (a →3 b))) = (a⊥ ∪ (a →3 b))
13		df-i3 46	. . . 4 (a →3 b) = (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b)))
14	13	lor 70	. . 3 (a⊥ ∪ (a →3 b)) = (a⊥ ∪ (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))))
15	12, 14	ax-r2 36	. 2 (a →3 (a →3 (a →3 b))) = (a⊥ ∪ (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))))
16		lem4 511	. 2 (a →3 (a →3 b)) = (a⊥ ∪ b)
17	11, 15, 16	3tr1 63	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:  i3abs2  523  i3th6  548