Theorem nom12 301

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

Assertion

Ref	Expression
nom12	(a ->2 (a ^ b)) = (a ->1 b)

Proof of Theorem nom12

Step	Hyp	Ref	Expression
1		oran 79	. . . . . . 7 (a v (a ^ b)) = (a_|_ ^ (a ^ b)_|_)_|_
2	1	ax-r1 34	. . . . . 6 (a_|_ ^ (a ^ b)_|_)_|_ = (a v (a ^ b))
3		a5b 112	. . . . . 6 (a v (a ^ b)) = a
4	2, 3	ax-r2 35	. . . . 5 (a_|_ ^ (a ^ b)_|_)_|_ = a
5	4	con3 65	. . . 4 (a_|_ ^ (a ^ b)_|_) = a_|_
6	5	lor 66	. . 3 ((a ^ b) v (a_|_ ^ (a ^ b)_|_)) = ((a ^ b) v a_|_)
7		ax-a2 30	. . 3 ((a ^ b) v a_|_) = (a_|_ v (a ^ b))
8	6, 7	ax-r2 35	. 2 ((a ^ b) v (a_|_ ^ (a ^ b)_|_)) = (a_|_ v (a ^ b))
9		df-i2 44	. 2 (a ->2 (a ^ b)) = ((a ^ b) v (a_|_ ^ (a ^ b)_|_))
10		df-i1 43	. 2 (a ->1 b) = (a_|_ v (a ^ b))
11	8, 9, 10	3tr1 60	1 (a ->2 (a ^ b)) = (a ->1 b)

Syntax hints:   = wb 1  _|_wn 4   v wo 6   ^ wa 7   ->1 wi1 13   ->2 wi2 14

This theorem is referenced by:  nom41 318

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

This theorem depends on definitions:  df-a 39  df-i1 43  df-i2 44

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