Theorem nom13 302

Description: Part of Lemma 3.3(14) from "Non-Orthomodular Models..." paper.

Assertion

Ref	Expression
nom13	(a ->3 (a ^ b)) = (a ->1 b)

Proof of Theorem nom13

Step	Hyp	Ref	Expression
1		oran 79	. . . . . . . . 9 (a v (a ^ b)) = (a_|_ ^ (a ^ b)_|_)_|_
2	1	ax-r1 34	. . . . . . . 8 (a_|_ ^ (a ^ b)_|_)_|_ = (a v (a ^ b))
3		a5b 112	. . . . . . . 8 (a v (a ^ b)) = a
4	2, 3	ax-r2 35	. . . . . . 7 (a_|_ ^ (a ^ b)_|_)_|_ = a
5	4	con3 65	. . . . . 6 (a_|_ ^ (a ^ b)_|_) = a_|_
6	5	lor 66	. . . . 5 ((a_|_ ^ (a ^ b)) v (a_|_ ^ (a ^ b)_|_)) = ((a_|_ ^ (a ^ b)) v a_|_)
7		lea 152	. . . . . 6 (a_|_ ^ (a ^ b)) =< a_|_
8	7	df-le2 123	. . . . 5 ((a_|_ ^ (a ^ b)) v a_|_) = a_|_
9	6, 8	ax-r2 35	. . . 4 ((a_|_ ^ (a ^ b)) v (a_|_ ^ (a ^ b)_|_)) = a_|_
10	9	ax-r5 37	. . 3 (((a_|_ ^ (a ^ b)) v (a_|_ ^ (a ^ b)_|_)) v (a ^ (a_|_ v (a ^ b)))) = (a_|_ v (a ^ (a_|_ v (a ^ b))))
11		womaa 214	. . 3 (a_|_ v (a ^ (a_|_ v (a ^ b)))) = (a_|_ v (a ^ b))
12	10, 11	ax-r2 35	. 2 (((a_|_ ^ (a ^ b)) v (a_|_ ^ (a ^ b)_|_)) v (a ^ (a_|_ v (a ^ b)))) = (a_|_ v (a ^ b))
13		df-i3 45	. 2 (a ->3 (a ^ b)) = (((a_|_ ^ (a ^ b)) v (a_|_ ^ (a ^ b)_|_)) v (a ^ (a_|_ v (a ^ b))))
14		df-i1 43	. 2 (a ->1 b) = (a_|_ v (a ^ b))
15	12, 13, 14	3tr1 60	1 (a ->3 (a ^ b)) = (a ->1 b)

Syntax hints:   = wb 1  _|_wn 4   v wo 6   ^ wa 7   ->1 wi1 13   ->3 wi3 15

This theorem is referenced by:  nom44 321

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37

This theorem depends on definitions:  df-a 39  df-t 40  df-f 41  df-i1 43  df-i3 45  df-le1 122  df-le2 123

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