Theorem nom25 310

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

Assertion

Ref	Expression
nom25	(a == (a ^ b)) = (a ->1 b)

Proof of Theorem nom25

Step	Hyp	Ref	Expression
1		anass 69	. . . . . 6 ((a ^ a) ^ b) = (a ^ (a ^ b))
2	1	ax-r1 34	. . . . 5 (a ^ (a ^ b)) = ((a ^ a) ^ b)
3		anidm 103	. . . . . 6 (a ^ a) = a
4	3	ran 71	. . . . 5 ((a ^ a) ^ b) = (a ^ b)
5	2, 4	ax-r2 35	. . . 4 (a ^ (a ^ b)) = (a ^ b)
6		oran3 85	. . . . . . 7 (a_|_ v b_|_) = (a ^ b)_|_
7	6	lan 70	. . . . . 6 (a_|_ ^ (a_|_ v b_|_)) = (a_|_ ^ (a ^ b)_|_)
8	7	ax-r1 34	. . . . 5 (a_|_ ^ (a ^ b)_|_) = (a_|_ ^ (a_|_ v b_|_))
9		a5c 113	. . . . 5 (a_|_ ^ (a_|_ v b_|_)) = a_|_
10	8, 9	ax-r2 35	. . . 4 (a_|_ ^ (a ^ b)_|_) = a_|_
11	5, 10	2or 67	. . 3 ((a ^ (a ^ b)) v (a_|_ ^ (a ^ b)_|_)) = ((a ^ b) v a_|_)
12		ax-a2 30	. . 3 ((a ^ b) v a_|_) = (a_|_ v (a ^ b))
13	11, 12	ax-r2 35	. 2 ((a ^ (a ^ b)) v (a_|_ ^ (a ^ b)_|_)) = (a_|_ v (a ^ b))
14		dfb 86	. 2 (a == (a ^ b)) = ((a ^ (a ^ b)) v (a_|_ ^ (a ^ b)_|_))
15		df-i1 43	. 2 (a ->1 b) = (a_|_ v (a ^ b))
16	13, 14, 15	3tr1 60	1 (a == (a ^ b)) = (a ->1 b)

Syntax hints:   = wb 1  _|_wn 4   == tb 5   v wo 6   ^ wa 7   ->1 wi1 13

This theorem is referenced by:  nom35 316  nom55 328

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-b 38  df-a 39  df-t 40  df-f 41  df-i1 43

metamath.org