Theorem ud3lem3c 556

Description: Lemma for unified disjunction.

Assertion

Ref	Expression
ud3lem3c	((a ->3 b)_|_ v (a v b)) = (a v b)

Proof of Theorem ud3lem3c

Step	Hyp	Ref	Expression
1		ud3lem0c 271	. . . 4 (a ->3 b)_|_ = (((a v b_|_) ^ (a v b)) ^ (a_|_ v (a ^ b_|_)))
2		an32 76	. . . . 5 (((a v b_|_) ^ (a v b)) ^ (a_|_ v (a ^ b_|_))) = (((a v b_|_) ^ (a_|_ v (a ^ b_|_))) ^ (a v b))
3		ancom 68	. . . . 5 (((a v b_|_) ^ (a_|_ v (a ^ b_|_))) ^ (a v b)) = ((a v b) ^ ((a v b_|_) ^ (a_|_ v (a ^ b_|_))))
4	2, 3	ax-r2 35	. . . 4 (((a v b_|_) ^ (a v b)) ^ (a_|_ v (a ^ b_|_))) = ((a v b) ^ ((a v b_|_) ^ (a_|_ v (a ^ b_|_))))
5	1, 4	ax-r2 35	. . 3 (a ->3 b)_|_ = ((a v b) ^ ((a v b_|_) ^ (a_|_ v (a ^ b_|_))))
6	5	ax-r5 37	. 2 ((a ->3 b)_|_ v (a v b)) = (((a v b) ^ ((a v b_|_) ^ (a_|_ v (a ^ b_|_)))) v (a v b))
7		ax-a2 30	. . 3 (((a v b) ^ ((a v b_|_) ^ (a_|_ v (a ^ b_|_)))) v (a v b)) = ((a v b) v ((a v b) ^ ((a v b_|_) ^ (a_|_ v (a ^ b_|_)))))
8		a5b 112	. . 3 ((a v b) v ((a v b) ^ ((a v b_|_) ^ (a_|_ v (a ^ b_|_))))) = (a v b)
9	7, 8	ax-r2 35	. 2 (((a v b) ^ ((a v b_|_) ^ (a_|_ v (a ^ b_|_)))) v (a v b)) = (a v b)
10	6, 9	ax-r2 35	1 ((a ->3 b)_|_ v (a v b)) = (a v b)

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

This theorem is referenced by:  ud3lem3d 557

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-i3 45

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