Theorem ud5lem3c 575

Description: Lemma for unified disjunction.

Assertion

Ref	Expression
ud5lem3c	((a ->5 b)_|_ ^ (a v (a_|_ ^ b))_|_) = (((a v b) ^ (a v b_|_)) ^ a_|_)

Proof of Theorem ud5lem3c

Step	Hyp	Ref	Expression
1		ud5lem0c 273	. . 3 (a ->5 b)_|_ = (((a_|_ v b_|_) ^ (a v b_|_)) ^ (a v b))
2		oran 79	. . . . 5 (a v (a_|_ ^ b)) = (a_|_ ^ (a_|_ ^ b)_|_)_|_
3	2	con2 64	. . . 4 (a v (a_|_ ^ b))_|_ = (a_|_ ^ (a_|_ ^ b)_|_)
4		anor2 81	. . . . . 6 (a_|_ ^ b) = (a v b_|_)_|_
5	4	con2 64	. . . . 5 (a_|_ ^ b)_|_ = (a v b_|_)
6	5	lan 70	. . . 4 (a_|_ ^ (a_|_ ^ b)_|_) = (a_|_ ^ (a v b_|_))
7	3, 6	ax-r2 35	. . 3 (a v (a_|_ ^ b))_|_ = (a_|_ ^ (a v b_|_))
8	1, 7	2an 72	. 2 ((a ->5 b)_|_ ^ (a v (a_|_ ^ b))_|_) = ((((a_|_ v b_|_) ^ (a v b_|_)) ^ (a v b)) ^ (a_|_ ^ (a v b_|_)))
9		an32 76	. . 3 ((((a_|_ v b_|_) ^ (a v b_|_)) ^ (a v b)) ^ (a_|_ ^ (a v b_|_))) = ((((a_|_ v b_|_) ^ (a v b_|_)) ^ (a_|_ ^ (a v b_|_))) ^ (a v b))
10		an4 78	. . . . . . 7 (((a_|_ v b_|_) ^ (a v b_|_)) ^ (a_|_ ^ (a v b_|_))) = (((a_|_ v b_|_) ^ a_|_) ^ ((a v b_|_) ^ (a v b_|_)))
11		ancom 68	. . . . . . . . . 10 ((a_|_ v b_|_) ^ a_|_) = (a_|_ ^ (a_|_ v b_|_))
12		a5c 113	. . . . . . . . . 10 (a_|_ ^ (a_|_ v b_|_)) = a_|_
13	11, 12	ax-r2 35	. . . . . . . . 9 ((a_|_ v b_|_) ^ a_|_) = a_|_
14		anidm 103	. . . . . . . . 9 ((a v b_|_) ^ (a v b_|_)) = (a v b_|_)
15	13, 14	2an 72	. . . . . . . 8 (((a_|_ v b_|_) ^ a_|_) ^ ((a v b_|_) ^ (a v b_|_))) = (a_|_ ^ (a v b_|_))
16		ancom 68	. . . . . . . 8 (a_|_ ^ (a v b_|_)) = ((a v b_|_) ^ a_|_)
17	15, 16	ax-r2 35	. . . . . . 7 (((a_|_ v b_|_) ^ a_|_) ^ ((a v b_|_) ^ (a v b_|_))) = ((a v b_|_) ^ a_|_)
18	10, 17	ax-r2 35	. . . . . 6 (((a_|_ v b_|_) ^ (a v b_|_)) ^ (a_|_ ^ (a v b_|_))) = ((a v b_|_) ^ a_|_)
19	18	ran 71	. . . . 5 ((((a_|_ v b_|_) ^ (a v b_|_)) ^ (a_|_ ^ (a v b_|_))) ^ (a v b)) = (((a v b_|_) ^ a_|_) ^ (a v b))
20		ancom 68	. . . . 5 (((a v b_|_) ^ a_|_) ^ (a v b)) = ((a v b) ^ ((a v b_|_) ^ a_|_))
21	19, 20	ax-r2 35	. . . 4 ((((a_|_ v b_|_) ^ (a v b_|_)) ^ (a_|_ ^ (a v b_|_))) ^ (a v b)) = ((a v b) ^ ((a v b_|_) ^ a_|_))
22		anass 69	. . . . 5 (((a v b) ^ (a v b_|_)) ^ a_|_) = ((a v b) ^ ((a v b_|_) ^ a_|_))
23	22	ax-r1 34	. . . 4 ((a v b) ^ ((a v b_|_) ^ a_|_)) = (((a v b) ^ (a v b_|_)) ^ a_|_)
24	21, 23	ax-r2 35	. . 3 ((((a_|_ v b_|_) ^ (a v b_|_)) ^ (a_|_ ^ (a v b_|_))) ^ (a v b)) = (((a v b) ^ (a v b_|_)) ^ a_|_)
25	9, 24	ax-r2 35	. 2 ((((a_|_ v b_|_) ^ (a v b_|_)) ^ (a v b)) ^ (a_|_ ^ (a v b_|_))) = (((a v b) ^ (a v b_|_)) ^ a_|_)
26	8, 25	ax-r2 35	1 ((a ->5 b)_|_ ^ (a v (a_|_ ^ b))_|_) = (((a v b) ^ (a v b_|_)) ^ a_|_)

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

This theorem is referenced by:  ud5lem3 576

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-a 39  df-t 40  df-f 41  df-i5 47

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