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_|_ v ((a_|_ ^ b) v (a_|_ ^ b_|_))) = ((a_|_ v (a_|_ ^ b)) v (a_|_ v (a_|_ ^ b_|_)))
2		a5b 112	. . . . . . 7 (a_|_ v (a_|_ ^ b)) = a_|_
3		a5b 112	. . . . . . 7 (a_|_ v (a_|_ ^ b_|_)) = a_|_
4	2, 3	2or 67	. . . . . 6 ((a_|_ v (a_|_ ^ b)) v (a_|_ v (a_|_ ^ b_|_))) = (a_|_ v a_|_)
5		oridm 102	. . . . . 6 (a_|_ v a_|_) = a_|_
6	4, 5	ax-r2 35	. . . . 5 ((a_|_ v (a_|_ ^ b)) v (a_|_ v (a_|_ ^ b_|_))) = a_|_
7	1, 6	ax-r2 35	. . . 4 (a_|_ v ((a_|_ ^ b) v (a_|_ ^ b_|_))) = a_|_
8	7	ax-r5 37	. . 3 ((a_|_ v ((a_|_ ^ b) v (a_|_ ^ b_|_))) v (a ^ (a_|_ v b))) = (a_|_ v (a ^ (a_|_ v b)))
9		ax-a3 31	. . 3 ((a_|_ v ((a_|_ ^ b) v (a_|_ ^ b_|_))) v (a ^ (a_|_ v b))) = (a_|_ v (((a_|_ ^ b) v (a_|_ ^ b_|_)) v (a ^ (a_|_ v b))))
10		omln 428	. . 3 (a_|_ v (a ^ (a_|_ v b))) = (a_|_ v b)
11	8, 9, 10	3tr2 61	. 2 (a_|_ v (((a_|_ ^ b) v (a_|_ ^ b_|_)) v (a ^ (a_|_ v b)))) = (a_|_ v b)
12		lem4 493	. . 3 (a ->3 (a ->3 (a ->3 b))) = (a_|_ v (a ->3 b))
13		df-i3 45	. . . 4 (a ->3 b) = (((a_|_ ^ b) v (a_|_ ^ b_|_)) v (a ^ (a_|_ v b)))
14	13	lor 66	. . 3 (a_|_ v (a ->3 b)) = (a_|_ v (((a_|_ ^ b) v (a_|_ ^ b_|_)) v (a ^ (a_|_ v b))))
15	12, 14	ax-r2 35	. 2 (a ->3 (a ->3 (a ->3 b))) = (a_|_ v (((a_|_ ^ b) v (a_|_ ^ b_|_)) v (a ^ (a_|_ v b))))
16		lem4 493	. 2 (a ->3 (a ->3 b)) = (a_|_ v 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   v 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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