Theorem nom40 317

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

Assertion

Ref	Expression
nom40	((a v b) ->0 b) = (a ->2 b)

Proof of Theorem nom40

Step	Hyp	Ref	Expression
1		nom10 299	. 2 (b_|_ ->0 (b_|_ ^ a_|_)) = (b_|_ ->1 a_|_)
2		ax-a2 30	. . . 4 ((a v b)_|_ v b) = (b v (a v b)_|_)
3		ax-a1 29	. . . . 5 b = b_|__|_
4		ancom 68	. . . . . . 7 (b_|_ ^ a_|_) = (a_|_ ^ b_|_)
5		anor3 82	. . . . . . 7 (a_|_ ^ b_|_) = (a v b)_|_
6	4, 5	ax-r2 35	. . . . . 6 (b_|_ ^ a_|_) = (a v b)_|_
7	6	ax-r1 34	. . . . 5 (a v b)_|_ = (b_|_ ^ a_|_)
8	3, 7	2or 67	. . . 4 (b v (a v b)_|_) = (b_|__|_ v (b_|_ ^ a_|_))
9	2, 8	ax-r2 35	. . 3 ((a v b)_|_ v b) = (b_|__|_ v (b_|_ ^ a_|_))
10		df-i0 42	. . 3 ((a v b) ->0 b) = ((a v b)_|_ v b)
11		df-i0 42	. . 3 (b_|_ ->0 (b_|_ ^ a_|_)) = (b_|__|_ v (b_|_ ^ a_|_))
12	9, 10, 11	3tr1 60	. 2 ((a v b) ->0 b) = (b_|_ ->0 (b_|_ ^ a_|_))
13		i2i1 259	. 2 (a ->2 b) = (b_|_ ->1 a_|_)
14	1, 12, 13	3tr1 60	1 ((a v b) ->0 b) = (a ->2 b)

Syntax hints:   = wb 1  _|_wn 4   v wo 6   ^ wa 7   ->0 wi0 12   ->1 wi1 13   ->2 wi2 14

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

This theorem depends on definitions:  df-a 39  df-i0 42  df-i1 43  df-i2 44

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