Theorem nom52 325

Description: Part of Lemma 3.3(15) from "Non-Orthomodular Models..." paper.

Assertion

Ref	Expression
nom52	((a v b) ==2 b) = (a ->2 b)

Proof of Theorem nom52

Step	Hyp	Ref	Expression
1		ancom 68	. . . . . . . . 9 (b_|_ ^ a_|_) = (a_|_ ^ b_|_)
2		anor3 82	. . . . . . . . 9 (a_|_ ^ b_|_) = (a v b)_|_
3	1, 2	ax-r2 35	. . . . . . . 8 (b_|_ ^ a_|_) = (a v b)_|_
4	3	ax-r1 34	. . . . . . 7 (a v b)_|_ = (b_|_ ^ a_|_)
5	4	ax-r4 36	. . . . . 6 (a v b)_|__|_ = (b_|_ ^ a_|_)_|_
6	5	lor 66	. . . . 5 (b_|_ v (a v b)_|__|_) = (b_|_ v (b_|_ ^ a_|_)_|_)
7	4	lan 70	. . . . . 6 (b_|_ ^ (a v b)_|_) = (b_|_ ^ (b_|_ ^ a_|_))
8	7	lor 66	. . . . 5 (b_|__|_ v (b_|_ ^ (a v b)_|_)) = (b_|__|_ v (b_|_ ^ (b_|_ ^ a_|_)))
9	6, 8	2an 72	. . . 4 ((b_|_ v (a v b)_|__|_) ^ (b_|__|_ v (b_|_ ^ (a v b)_|_))) = ((b_|_ v (b_|_ ^ a_|_)_|_) ^ (b_|__|_ v (b_|_ ^ (b_|_ ^ a_|_))))
10		df-id1 49	. . . 4 (b_|_ ==1 (a v b)_|_) = ((b_|_ v (a v b)_|__|_) ^ (b_|__|_ v (b_|_ ^ (a v b)_|_)))
11		df-id1 49	. . . 4 (b_|_ ==1 (b_|_ ^ a_|_)) = ((b_|_ v (b_|_ ^ a_|_)_|_) ^ (b_|__|_ v (b_|_ ^ (b_|_ ^ a_|_))))
12	9, 10, 11	3tr1 60	. . 3 (b_|_ ==1 (a v b)_|_) = (b_|_ ==1 (b_|_ ^ a_|_))
13		nom21 306	. . 3 (b_|_ ==1 (b_|_ ^ a_|_)) = (b_|_ ->1 a_|_)
14	12, 13	ax-r2 35	. 2 (b_|_ ==1 (a v b)_|_) = (b_|_ ->1 a_|_)
15		nomcon2 295	. 2 ((a v b) ==2 b) = (b_|_ ==1 (a v b)_|_)
16		i2i1 259	. 2 (a ->2 b) = (b_|_ ->1 a_|_)
17	14, 15, 16	3tr1 60	1 ((a v b) ==2 b) = (a ->2 b)

Syntax hints:   = wb 1  _|_wn 4   v wo 6   ^ wa 7   ->1 wi1 13   ->2 wi2 14   ==1 wid1 19   ==2 wid2 20

This theorem is referenced by:  nom63 332

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-i2 44  df-id1 49  df-id2 50  df-le1 122  df-le2 123

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