Theorem nom54 327

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

Assertion

Ref	Expression
nom54	((a v b) ==4 b) = (a ->2 b)

Proof of Theorem nom54

Step	Hyp	Ref	Expression
1		ancom 68	. . . . . . . 8 (b_|_ ^ a_|_) = (a_|_ ^ b_|_)
2		anor3 82	. . . . . . . 8 (a_|_ ^ b_|_) = (a v b)_|_
3	1, 2	ax-r2 35	. . . . . . 7 (b_|_ ^ a_|_) = (a v b)_|_
4	3	lor 66	. . . . . 6 (b_|__|_ v (b_|_ ^ a_|_)) = (b_|__|_ v (a v b)_|_)
5	3	ax-r4 36	. . . . . . . 8 (b_|_ ^ a_|_)_|_ = (a v b)_|__|_
6	5	lan 70	. . . . . . 7 (b_|__|_ ^ (b_|_ ^ a_|_)_|_) = (b_|__|_ ^ (a v b)_|__|_)
7	6	lor 66	. . . . . 6 (b_|_ v (b_|__|_ ^ (b_|_ ^ a_|_)_|_)) = (b_|_ v (b_|__|_ ^ (a v b)_|__|_))
8	4, 7	2an 72	. . . . 5 ((b_|__|_ v (b_|_ ^ a_|_)) ^ (b_|_ v (b_|__|_ ^ (b_|_ ^ a_|_)_|_))) = ((b_|__|_ v (a v b)_|_) ^ (b_|_ v (b_|__|_ ^ (a v b)_|__|_)))
9		df-id3 51	. . . . 5 (b_|_ ==3 (b_|_ ^ a_|_)) = ((b_|__|_ v (b_|_ ^ a_|_)) ^ (b_|_ v (b_|__|_ ^ (b_|_ ^ a_|_)_|_)))
10		df-id3 51	. . . . 5 (b_|_ ==3 (a v b)_|_) = ((b_|__|_ v (a v b)_|_) ^ (b_|_ v (b_|__|_ ^ (a v b)_|__|_)))
11	8, 9, 10	3tr1 60	. . . 4 (b_|_ ==3 (b_|_ ^ a_|_)) = (b_|_ ==3 (a v b)_|_)
12	11	ax-r1 34	. . 3 (b_|_ ==3 (a v b)_|_) = (b_|_ ==3 (b_|_ ^ a_|_))
13		nom23 308	. . 3 (b_|_ ==3 (b_|_ ^ a_|_)) = (b_|_ ->1 a_|_)
14	12, 13	ax-r2 35	. 2 (b_|_ ==3 (a v b)_|_) = (b_|_ ->1 a_|_)
15		nomcon4 297	. 2 ((a v b) ==4 b) = (b_|_ ==3 (a v b)_|_)
16		i2i1 259	. 2 (a ->2 b) = (b_|_ ->1 a_|_)
17	14, 15, 16	3tr1 60	1 ((a v b) ==4 b) = (a ->2 b)

Syntax hints:   = wb 1  _|_wn 4   v wo 6   ^ wa 7   ->1 wi1 13   ->2 wi2 14   ==3 wid3 21   ==4 wid4 22

This theorem is referenced by:  nom61 330

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a4 32  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-i1 43  df-i2 44  df-id1 49  df-id2 50  df-id3 51  df-id4 52  df-le1 122  df-le2 123

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