Theorem u5lemnab 636

Description: Lemma for relevance implication study.

Assertion

Ref	Expression
u5lemnab	((a ->5 b)_|_ ^ b) = (((a v b_|_) ^ (a_|_ v b_|_)) ^ b)

Proof of Theorem u5lemnab

Step	Hyp	Ref	Expression
1		u5lemonb 621	. . . 4 ((a ->5 b) v b_|_) = (((a ^ b) v (a_|_ ^ b)) v b_|_)
2		ax-a2 30	. . . . . 6 ((a ^ b) v (a_|_ ^ b)) = ((a_|_ ^ b) v (a ^ b))
3		anor2 81	. . . . . . . 8 (a_|_ ^ b) = (a v b_|_)_|_
4		df-a 39	. . . . . . . 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	ax-r5 37	. . . 4 (((a ^ b) v (a_|_ ^ b)) v b_|_) = (((a v b_|_) ^ (a_|_ v b_|_))_|_ v b_|_)
10	1, 9	ax-r2 35	. . 3 ((a ->5 b) v b_|_) = (((a v b_|_) ^ (a_|_ v b_|_))_|_ v b_|_)
11		oran1 83	. . 3 ((a ->5 b) v b_|_) = ((a ->5 b)_|_ ^ b)_|_
12		oran3 85	. . 3 (((a v b_|_) ^ (a_|_ v b_|_))_|_ v b_|_) = (((a v b_|_) ^ (a_|_ v b_|_)) ^ b)_|_
13	10, 11, 12	3tr2 61	. 2 ((a ->5 b)_|_ ^ b)_|_ = (((a v b_|_) ^ (a_|_ v b_|_)) ^ b)_|_
14	13	con1 63	1 ((a ->5 b)_|_ ^ b) = (((a v b_|_) ^ (a_|_ v b_|_)) ^ 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  df-le1 122  df-le2 123

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