Theorem u1lemoa 602

Description: Lemma for Sasaki implication study.

Assertion

Ref	Expression
u1lemoa	((a ->1 b) v a) = 1

Proof of Theorem u1lemoa

Step	Hyp	Ref	Expression
1		df-i1 43	. . 3 (a ->1 b) = (a_|_ v (a ^ b))
2	1	ax-r5 37	. 2 ((a ->1 b) v a) = ((a_|_ v (a ^ b)) v a)
3		ax-a2 30	. . 3 ((a_|_ v (a ^ b)) v a) = (a v (a_|_ v (a ^ b)))
4		ax-a3 31	. . . . 5 ((a v a_|_) v (a ^ b)) = (a v (a_|_ v (a ^ b)))
5	4	ax-r1 34	. . . 4 (a v (a_|_ v (a ^ b))) = ((a v a_|_) v (a ^ b))
6		ax-a2 30	. . . . 5 ((a v a_|_) v (a ^ b)) = ((a ^ b) v (a v a_|_))
7		df-t 40	. . . . . . . 8 1 = (a v a_|_)
8	7	lor 66	. . . . . . 7 ((a ^ b) v 1) = ((a ^ b) v (a v a_|_))
9	8	ax-r1 34	. . . . . 6 ((a ^ b) v (a v a_|_)) = ((a ^ b) v 1)
10		or1 96	. . . . . 6 ((a ^ b) v 1) = 1
11	9, 10	ax-r2 35	. . . . 5 ((a ^ b) v (a v a_|_)) = 1
12	6, 11	ax-r2 35	. . . 4 ((a v a_|_) v (a ^ b)) = 1
13	5, 12	ax-r2 35	. . 3 (a v (a_|_ v (a ^ b))) = 1
14	3, 13	ax-r2 35	. 2 ((a_|_ v (a ^ b)) v a) = 1
15	2, 14	ax-r2 35	1 ((a ->1 b) v a) = 1

Syntax hints:   = wb 1  _|_wn 4   v wo 6   ^ wa 7  1wt 9   ->1 wi1 13

This theorem is referenced by:  u1lemnana 627  oi3oa3lem1 714  i1orni1 829  oau 909

This theorem was proved from axioms:  ax-a2 30  ax-a3 31  ax-a4 32  ax-r1 34  ax-r2 35  ax-r5 37

This theorem depends on definitions:  df-t 40  df-i1 43

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