Theorem ud5lem2 572

Description: Lemma for unified disjunction.

Assertion

Ref	Expression
ud5lem2	((a v b_|_) ->5 a) = (a v (a_|_ ^ b))

Proof of Theorem ud5lem2

Step	Hyp	Ref	Expression
1		df-i5 47	. 2 ((a v b_|_) ->5 a) = ((((a v b_|_) ^ a) v ((a v b_|_)_|_ ^ a)) v ((a v b_|_)_|_ ^ a_|_))
2		ax-a3 31	. . 3 ((((a v b_|_) ^ a) v ((a v b_|_)_|_ ^ a)) v ((a v b_|_)_|_ ^ a_|_)) = (((a v b_|_) ^ a) v (((a v b_|_)_|_ ^ a) v ((a v b_|_)_|_ ^ a_|_)))
3		ancom 68	. . . . 5 ((a v b_|_) ^ a) = (a ^ (a v b_|_))
4		a5c 113	. . . . 5 (a ^ (a v b_|_)) = a
5	3, 4	ax-r2 35	. . . 4 ((a v b_|_) ^ a) = a
6		ax-a2 30	. . . . 5 (((a v b_|_)_|_ ^ a) v ((a v b_|_)_|_ ^ a_|_)) = (((a v b_|_)_|_ ^ a_|_) v ((a v b_|_)_|_ ^ a))
7		anor2 81	. . . . . . . . . 10 (a_|_ ^ b) = (a v b_|_)_|_
8	7	ax-r1 34	. . . . . . . . 9 (a v b_|_)_|_ = (a_|_ ^ b)
9	8	ran 71	. . . . . . . 8 ((a v b_|_)_|_ ^ a_|_) = ((a_|_ ^ b) ^ a_|_)
10		an32 76	. . . . . . . . 9 ((a_|_ ^ b) ^ a_|_) = ((a_|_ ^ a_|_) ^ b)
11		anidm 103	. . . . . . . . . 10 (a_|_ ^ a_|_) = a_|_
12	11	ran 71	. . . . . . . . 9 ((a_|_ ^ a_|_) ^ b) = (a_|_ ^ b)
13	10, 12	ax-r2 35	. . . . . . . 8 ((a_|_ ^ b) ^ a_|_) = (a_|_ ^ b)
14	9, 13	ax-r2 35	. . . . . . 7 ((a v b_|_)_|_ ^ a_|_) = (a_|_ ^ b)
15	8	ran 71	. . . . . . . 8 ((a v b_|_)_|_ ^ a) = ((a_|_ ^ b) ^ a)
16		an32 76	. . . . . . . . 9 ((a_|_ ^ b) ^ a) = ((a_|_ ^ a) ^ b)
17		ancom 68	. . . . . . . . . 10 ((a_|_ ^ a) ^ b) = (b ^ (a_|_ ^ a))
18		ancom 68	. . . . . . . . . . . . 13 (a_|_ ^ a) = (a ^ a_|_)
19		dff 93	. . . . . . . . . . . . . 14 0 = (a ^ a_|_)
20	19	ax-r1 34	. . . . . . . . . . . . 13 (a ^ a_|_) = 0
21	18, 20	ax-r2 35	. . . . . . . . . . . 12 (a_|_ ^ a) = 0
22	21	lan 70	. . . . . . . . . . 11 (b ^ (a_|_ ^ a)) = (b ^ 0)
23		an0 100	. . . . . . . . . . 11 (b ^ 0) = 0
24	22, 23	ax-r2 35	. . . . . . . . . 10 (b ^ (a_|_ ^ a)) = 0
25	17, 24	ax-r2 35	. . . . . . . . 9 ((a_|_ ^ a) ^ b) = 0
26	16, 25	ax-r2 35	. . . . . . . 8 ((a_|_ ^ b) ^ a) = 0
27	15, 26	ax-r2 35	. . . . . . 7 ((a v b_|_)_|_ ^ a) = 0
28	14, 27	2or 67	. . . . . 6 (((a v b_|_)_|_ ^ a_|_) v ((a v b_|_)_|_ ^ a)) = ((a_|_ ^ b) v 0)
29		or0 94	. . . . . 6 ((a_|_ ^ b) v 0) = (a_|_ ^ b)
30	28, 29	ax-r2 35	. . . . 5 (((a v b_|_)_|_ ^ a_|_) v ((a v b_|_)_|_ ^ a)) = (a_|_ ^ b)
31	6, 30	ax-r2 35	. . . 4 (((a v b_|_)_|_ ^ a) v ((a v b_|_)_|_ ^ a_|_)) = (a_|_ ^ b)
32	5, 31	2or 67	. . 3 (((a v b_|_) ^ a) v (((a v b_|_)_|_ ^ a) v ((a v b_|_)_|_ ^ a_|_))) = (a v (a_|_ ^ b))
33	2, 32	ax-r2 35	. 2 ((((a v b_|_) ^ a) v ((a v b_|_)_|_ ^ a)) v ((a v b_|_)_|_ ^ a_|_)) = (a v (a_|_ ^ b))
34	1, 33	ax-r2 35	1 ((a v b_|_) ->5 a) = (a v (a_|_ ^ b))

Syntax hints:   = wb 1  _|_wn 4   v wo 6   ^ wa 7  0wf 10   ->5 wi5 17

This theorem is referenced by:  ud5 581

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  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-f 41  df-i5 47

metamath.org