Theorem u2lemaa 583

Description: Lemma for Dishkant implication study.

Assertion

Ref	Expression
u2lemaa	((a ->2 b) ^ a) = (a ^ b)

Proof of Theorem u2lemaa

Step	Hyp	Ref	Expression
1		df-i2 44	. . 3 (a ->2 b) = (b v (a_|_ ^ b_|_))
2	1	ran 71	. 2 ((a ->2 b) ^ a) = ((b v (a_|_ ^ b_|_)) ^ a)
3		ax-a2 30	. . . 4 (b v (a_|_ ^ b_|_)) = ((a_|_ ^ b_|_) v b)
4	3	ran 71	. . 3 ((b v (a_|_ ^ b_|_)) ^ a) = (((a_|_ ^ b_|_) v b) ^ a)
5		coman1 177	. . . . . 6 (a_|_ ^ b_|_) C a_|_
6	5	comcom7 442	. . . . 5 (a_|_ ^ b_|_) C a
7		coman2 178	. . . . . 6 (a_|_ ^ b_|_) C b_|_
8	7	comcom7 442	. . . . 5 (a_|_ ^ b_|_) C b
9	6, 8	fh2r 456	. . . 4 (((a_|_ ^ b_|_) v b) ^ a) = (((a_|_ ^ b_|_) ^ a) v (b ^ a))
10		ax-a2 30	. . . . 5 (((a_|_ ^ b_|_) ^ a) v (b ^ a)) = ((b ^ a) v ((a_|_ ^ b_|_) ^ a))
11		ancom 68	. . . . . . 7 (b ^ a) = (a ^ b)
12		ancom 68	. . . . . . . 8 ((a_|_ ^ b_|_) ^ a) = (a ^ (a_|_ ^ b_|_))
13		anass 69	. . . . . . . . . 10 ((a ^ a_|_) ^ b_|_) = (a ^ (a_|_ ^ b_|_))
14	13	ax-r1 34	. . . . . . . . 9 (a ^ (a_|_ ^ b_|_)) = ((a ^ a_|_) ^ b_|_)
15		ancom 68	. . . . . . . . . 10 ((a ^ a_|_) ^ b_|_) = (b_|_ ^ (a ^ a_|_))
16		dff 93	. . . . . . . . . . . . 13 0 = (a ^ a_|_)
17	16	ax-r1 34	. . . . . . . . . . . 12 (a ^ a_|_) = 0
18	17	lan 70	. . . . . . . . . . 11 (b_|_ ^ (a ^ a_|_)) = (b_|_ ^ 0)
19		an0 100	. . . . . . . . . . 11 (b_|_ ^ 0) = 0
20	18, 19	ax-r2 35	. . . . . . . . . 10 (b_|_ ^ (a ^ a_|_)) = 0
21	15, 20	ax-r2 35	. . . . . . . . 9 ((a ^ a_|_) ^ b_|_) = 0
22	14, 21	ax-r2 35	. . . . . . . 8 (a ^ (a_|_ ^ b_|_)) = 0
23	12, 22	ax-r2 35	. . . . . . 7 ((a_|_ ^ b_|_) ^ a) = 0
24	11, 23	2or 67	. . . . . 6 ((b ^ a) v ((a_|_ ^ b_|_) ^ a)) = ((a ^ b) v 0)
25		or0 94	. . . . . 6 ((a ^ b) v 0) = (a ^ b)
26	24, 25	ax-r2 35	. . . . 5 ((b ^ a) v ((a_|_ ^ b_|_) ^ a)) = (a ^ b)
27	10, 26	ax-r2 35	. . . 4 (((a_|_ ^ b_|_) ^ a) v (b ^ a)) = (a ^ b)
28	9, 27	ax-r2 35	. . 3 (((a_|_ ^ b_|_) v b) ^ a) = (a ^ b)
29	4, 28	ax-r2 35	. 2 ((b v (a_|_ ^ b_|_)) ^ a) = (a ^ b)
30	2, 29	ax-r2 35	1 ((a ->2 b) ^ a) = (a ^ b)

Syntax hints:   = wb 1  _|_wn 4   v wo 6   ^ wa 7  0wf 10   ->2 wi2 14

This theorem is referenced by:  u2lemnona 648

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-i2 44  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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