Theorem ud3lem3b 555

Description: Lemma for unified disjunction.

Assertion

Ref	Expression
ud3lem3b	((a ->3 b)_|_ ^ (a v b)_|_) = 0

Proof of Theorem ud3lem3b

Step	Hyp	Ref	Expression
1		ud3lem0c 271	. . 3 (a ->3 b)_|_ = (((a v b_|_) ^ (a v b)) ^ (a_|_ v (a ^ b_|_)))
2	1	ran 71	. 2 ((a ->3 b)_|_ ^ (a v b)_|_) = ((((a v b_|_) ^ (a v b)) ^ (a_|_ v (a ^ b_|_))) ^ (a v b)_|_)
3		an32 76	. . 3 ((((a v b_|_) ^ (a v b)) ^ (a_|_ v (a ^ b_|_))) ^ (a v b)_|_) = ((((a v b_|_) ^ (a v b)) ^ (a v b)_|_) ^ (a_|_ v (a ^ b_|_)))
4		anass 69	. . . . . 6 (((a v b_|_) ^ (a v b)) ^ (a v b)_|_) = ((a v b_|_) ^ ((a v b) ^ (a v b)_|_))
5		dff 93	. . . . . . . . 9 0 = ((a v b) ^ (a v b)_|_)
6	5	ax-r1 34	. . . . . . . 8 ((a v b) ^ (a v b)_|_) = 0
7	6	lan 70	. . . . . . 7 ((a v b_|_) ^ ((a v b) ^ (a v b)_|_)) = ((a v b_|_) ^ 0)
8		an0 100	. . . . . . 7 ((a v b_|_) ^ 0) = 0
9	7, 8	ax-r2 35	. . . . . 6 ((a v b_|_) ^ ((a v b) ^ (a v b)_|_)) = 0
10	4, 9	ax-r2 35	. . . . 5 (((a v b_|_) ^ (a v b)) ^ (a v b)_|_) = 0
11	10	ran 71	. . . 4 ((((a v b_|_) ^ (a v b)) ^ (a v b)_|_) ^ (a_|_ v (a ^ b_|_))) = (0 ^ (a_|_ v (a ^ b_|_)))
12		an0r 101	. . . 4 (0 ^ (a_|_ v (a ^ b_|_))) = 0
13	11, 12	ax-r2 35	. . 3 ((((a v b_|_) ^ (a v b)) ^ (a v b)_|_) ^ (a_|_ v (a ^ b_|_))) = 0
14	3, 13	ax-r2 35	. 2 ((((a v b_|_) ^ (a v b)) ^ (a_|_ v (a ^ b_|_))) ^ (a v b)_|_) = 0
15	2, 14	ax-r2 35	1 ((a ->3 b)_|_ ^ (a v b)_|_) = 0

Syntax hints:   = wb 1  _|_wn 4   v wo 6   ^ wa 7  0wf 10   ->3 wi3 15

This theorem is referenced by:  ud3lem3 558

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a4 32  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-i3 45

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