Theorem u5lem5 747

Description: Lemma for unified implication study.

Assertion

Ref	Expression
u5lem5	(a ->5 (a ->5 b)) = (a_|_ v (a ^ b))

Proof of Theorem u5lem5

Step	Hyp	Ref	Expression
1		df-i5 47	. 2 (a ->5 (a ->5 b)) = (((a ^ (a ->5 b)) v (a_|_ ^ (a ->5 b))) v (a_|_ ^ (a ->5 b)_|_))
2		u5lemc1 666	. . . . . . . 8 a C (a ->5 b)
3	2	comcom 435	. . . . . . 7 (a ->5 b) C a
4	3	comcom2 175	. . . . . . 7 (a ->5 b) C a_|_
5	3, 4	fh1r 455	. . . . . 6 ((a v a_|_) ^ (a ->5 b)) = ((a ^ (a ->5 b)) v (a_|_ ^ (a ->5 b)))
6	5	ax-r1 34	. . . . 5 ((a ^ (a ->5 b)) v (a_|_ ^ (a ->5 b))) = ((a v a_|_) ^ (a ->5 b))
7		ancom 68	. . . . . 6 ((a v a_|_) ^ (a ->5 b)) = ((a ->5 b) ^ (a v a_|_))
8		df-t 40	. . . . . . . . 9 1 = (a v a_|_)
9	8	ax-r1 34	. . . . . . . 8 (a v a_|_) = 1
10	9	lan 70	. . . . . . 7 ((a ->5 b) ^ (a v a_|_)) = ((a ->5 b) ^ 1)
11		an1 98	. . . . . . 7 ((a ->5 b) ^ 1) = (a ->5 b)
12	10, 11	ax-r2 35	. . . . . 6 ((a ->5 b) ^ (a v a_|_)) = (a ->5 b)
13	7, 12	ax-r2 35	. . . . 5 ((a v a_|_) ^ (a ->5 b)) = (a ->5 b)
14	6, 13	ax-r2 35	. . . 4 ((a ^ (a ->5 b)) v (a_|_ ^ (a ->5 b))) = (a ->5 b)
15	14	ax-r5 37	. . 3 (((a ^ (a ->5 b)) v (a_|_ ^ (a ->5 b))) v (a_|_ ^ (a ->5 b)_|_)) = ((a ->5 b) v (a_|_ ^ (a ->5 b)_|_))
16	2	comcom3 436	. . . . 5 a_|_ C (a ->5 b)
17	2	comcom4 437	. . . . 5 a_|_ C (a ->5 b)_|_
18	16, 17	fh4 454	. . . 4 ((a ->5 b) v (a_|_ ^ (a ->5 b)_|_)) = (((a ->5 b) v a_|_) ^ ((a ->5 b) v (a ->5 b)_|_))
19		df-t 40	. . . . . . 7 1 = ((a ->5 b) v (a ->5 b)_|_)
20	19	ax-r1 34	. . . . . 6 ((a ->5 b) v (a ->5 b)_|_) = 1
21	20	lan 70	. . . . 5 (((a ->5 b) v a_|_) ^ ((a ->5 b) v (a ->5 b)_|_)) = (((a ->5 b) v a_|_) ^ 1)
22		an1 98	. . . . . 6 (((a ->5 b) v a_|_) ^ 1) = ((a ->5 b) v a_|_)
23		u5lemona 611	. . . . . 6 ((a ->5 b) v a_|_) = (a_|_ v (a ^ b))
24	22, 23	ax-r2 35	. . . . 5 (((a ->5 b) v a_|_) ^ 1) = (a_|_ v (a ^ b))
25	21, 24	ax-r2 35	. . . 4 (((a ->5 b) v a_|_) ^ ((a ->5 b) v (a ->5 b)_|_)) = (a_|_ v (a ^ b))
26	18, 25	ax-r2 35	. . 3 ((a ->5 b) v (a_|_ ^ (a ->5 b)_|_)) = (a_|_ v (a ^ b))
27	15, 26	ax-r2 35	. 2 (((a ^ (a ->5 b)) v (a_|_ ^ (a ->5 b))) v (a_|_ ^ (a ->5 b)_|_)) = (a_|_ v (a ^ b))
28	1, 27	ax-r2 35	1 (a ->5 (a ->5 b)) = (a_|_ v (a ^ b))

Syntax hints:   = wb 1  _|_wn 4   v wo 6   ^ wa 7  1wt 9   ->5 wi5 17

This theorem is referenced by:  u5lem6 751

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  ax-r3 421

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-i5 47  df-le1 122  df-le2 123  df-c1 124  df-c2 125

metamath.org