Theorem ud5lem0c 273

Description: Lemma for unified disjunction.

Assertion

Ref	Expression
ud5lem0c	(a ->5 b)_|_ = (((a_|_ v b_|_) ^ (a v b_|_)) ^ (a v b))

Proof of Theorem ud5lem0c

Step	Hyp	Ref	Expression
1		df-i5 47	. . 3 (a ->5 b) = (((a ^ b) v (a_|_ ^ b)) v (a_|_ ^ b_|_))
2		oran 79	. . . 4 (((a ^ b) v (a_|_ ^ b)) v (a_|_ ^ b_|_)) = (((a ^ b) v (a_|_ ^ b))_|_ ^ (a_|_ ^ b_|_)_|_)_|_
3		oran 79	. . . . . . . 8 ((a ^ b) v (a_|_ ^ b)) = ((a ^ b)_|_ ^ (a_|_ ^ b)_|_)_|_
4		df-a 39	. . . . . . . . . . 11 (a ^ b) = (a_|_ v b_|_)_|_
5	4	con2 64	. . . . . . . . . 10 (a ^ b)_|_ = (a_|_ v b_|_)
6		anor2 81	. . . . . . . . . . 11 (a_|_ ^ b) = (a v b_|_)_|_
7	6	con2 64	. . . . . . . . . 10 (a_|_ ^ b)_|_ = (a v b_|_)
8	5, 7	2an 72	. . . . . . . . 9 ((a ^ b)_|_ ^ (a_|_ ^ b)_|_) = ((a_|_ v b_|_) ^ (a v b_|_))
9	8	ax-r4 36	. . . . . . . 8 ((a ^ b)_|_ ^ (a_|_ ^ b)_|_)_|_ = ((a_|_ v b_|_) ^ (a v b_|_))_|_
10	3, 9	ax-r2 35	. . . . . . 7 ((a ^ b) v (a_|_ ^ b)) = ((a_|_ v b_|_) ^ (a v b_|_))_|_
11	10	con2 64	. . . . . 6 ((a ^ b) v (a_|_ ^ b))_|_ = ((a_|_ v b_|_) ^ (a v b_|_))
12		oran 79	. . . . . . 7 (a v b) = (a_|_ ^ b_|_)_|_
13	12	ax-r1 34	. . . . . 6 (a_|_ ^ b_|_)_|_ = (a v b)
14	11, 13	2an 72	. . . . 5 (((a ^ b) v (a_|_ ^ b))_|_ ^ (a_|_ ^ b_|_)_|_) = (((a_|_ v b_|_) ^ (a v b_|_)) ^ (a v b))
15	14	ax-r4 36	. . . 4 (((a ^ b) v (a_|_ ^ b))_|_ ^ (a_|_ ^ b_|_)_|_)_|_ = (((a_|_ v b_|_) ^ (a v b_|_)) ^ (a v b))_|_
16	2, 15	ax-r2 35	. . 3 (((a ^ b) v (a_|_ ^ b)) v (a_|_ ^ b_|_)) = (((a_|_ v b_|_) ^ (a v b_|_)) ^ (a v b))_|_
17	1, 16	ax-r2 35	. 2 (a ->5 b) = (((a_|_ v b_|_) ^ (a v b_|_)) ^ (a v b))_|_
18	17	con2 64	1 (a ->5 b)_|_ = (((a_|_ v b_|_) ^ (a v b_|_)) ^ (a v b))

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

This theorem is referenced by:  ud5lem1b 569  ud5lem1c 570  ud5lem3b 574  ud5lem3c 575

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37

This theorem depends on definitions:  df-a 39  df-i5 47

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