Theorem u5lemnana 631

Description: Lemma for relevance implication study.

Assertion

Ref	Expression
u5lemnana	((a ->5 b)_|_ ^ a_|_) = (a_|_ ^ ((a v b) ^ (a v b_|_)))

Proof of Theorem u5lemnana

Step	Hyp	Ref	Expression
1		u5lemoa 606	. . . 4 ((a ->5 b) v a) = (a v ((a_|_ ^ b) v (a_|_ ^ b_|_)))
2		ax-a2 30	. . . . . 6 ((a_|_ ^ b) v (a_|_ ^ b_|_)) = ((a_|_ ^ b_|_) v (a_|_ ^ b))
3		anor3 82	. . . . . . . 8 (a_|_ ^ b_|_) = (a v b)_|_
4		anor2 81	. . . . . . . 8 (a_|_ ^ b) = (a v b_|_)_|_
5	3, 4	2or 67	. . . . . . 7 ((a_|_ ^ b_|_) v (a_|_ ^ b)) = ((a v b)_|_ v (a v b_|_)_|_)
6		oran3 85	. . . . . . 7 ((a v b)_|_ v (a v b_|_)_|_) = ((a v b) ^ (a v b_|_))_|_
7	5, 6	ax-r2 35	. . . . . 6 ((a_|_ ^ b_|_) v (a_|_ ^ b)) = ((a v b) ^ (a v b_|_))_|_
8	2, 7	ax-r2 35	. . . . 5 ((a_|_ ^ b) v (a_|_ ^ b_|_)) = ((a v b) ^ (a v b_|_))_|_
9	8	lor 66	. . . 4 (a v ((a_|_ ^ b) v (a_|_ ^ b_|_))) = (a v ((a v b) ^ (a v b_|_))_|_)
10	1, 9	ax-r2 35	. . 3 ((a ->5 b) v a) = (a v ((a v b) ^ (a v b_|_))_|_)
11		oran 79	. . 3 ((a ->5 b) v a) = ((a ->5 b)_|_ ^ a_|_)_|_
12		oran1 83	. . 3 (a v ((a v b) ^ (a v b_|_))_|_) = (a_|_ ^ ((a v b) ^ (a v b_|_)))_|_
13	10, 11, 12	3tr2 61	. 2 ((a ->5 b)_|_ ^ a_|_)_|_ = (a_|_ ^ ((a v b) ^ (a v b_|_)))_|_
14	13	con1 63	1 ((a ->5 b)_|_ ^ a_|_) = (a_|_ ^ ((a v b) ^ (a v b_|_)))

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

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-i5 47

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