Theorem i3abs1 504

Description: Antecedent absorption.

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 104	. . . . 5 (a⊥ ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) = ((a⊥ ∪ (a⊥ ∩ b)) ∪ (a⊥ ∪ (a⊥ ∩ b⊥ )))
2		a5b 112	. . . . . . 7 (a⊥ ∪ (a⊥ ∩ b)) = a⊥
3		a5b 112	. . . . . . 7 (a⊥ ∪ (a⊥ ∩ b⊥ )) = a⊥
4	2, 3	2or 67	. . . . . 6 ((a⊥ ∪ (a⊥ ∩ b)) ∪ (a⊥ ∪ (a⊥ ∩ b⊥ ))) = (a⊥ ∪ a⊥ )
5		oridm 102	. . . . . 6 (a⊥ ∪ a⊥ ) = a⊥
6	4, 5	ax-r2 35	. . . . 5 ((a⊥ ∪ (a⊥ ∩ b)) ∪ (a⊥ ∪ (a⊥ ∩ b⊥ ))) = a⊥
7	1, 6	ax-r2 35	. . . 4 (a⊥ ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) = a⊥
8	7	ax-r5 37	. . 3 ((a⊥ ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) ∪ (a ∩ (a⊥ ∪ b))) = (a⊥ ∪ (a ∩ (a⊥ ∪ b)))
9		ax-a3 31	. . 3 ((a⊥ ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) ∪ (a ∩ (a⊥ ∪ b))) = (a⊥ ∪ (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))))
10		omln 428	. . 3 (a⊥ ∪ (a ∩ (a⊥ ∪ b))) = (a⊥ ∪ b)
11	8, 9, 10	3tr2 61	. 2 (a⊥ ∪ (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b)))) = (a⊥ ∪ b)
12		lem4 493	. . 3 (a →3 (a →3 (a →3 b))) = (a⊥ ∪ (a →3 b))
13		df-i3 45	. . . 4 (a →3 b) = (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b)))
14	13	lor 66	. . 3 (a⊥ ∪ (a →3 b)) = (a⊥ ∪ (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))))
15	12, 14	ax-r2 35	. 2 (a →3 (a →3 (a →3 b))) = (a⊥ ∪ (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))))
16		lem4 493	. 2 (a →3 (a →3 b)) = (a⊥ ∪ b)
17	11, 15, 16	3tr1 60	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 15

This theorem is referenced by:  i3abs2 505  i3th6 530

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

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