Theorem u5lemnanb 641

Description: Lemma for relevance implication study.

Assertion

Ref	Expression
u5lemnanb	((a ->5 b)_|_ ^ b_|_) = ((a v b) ^ b_|_)

Proof of Theorem u5lemnanb

Step	Hyp	Ref	Expression
1		u5lemob 616	. . . 4 ((a ->5 b) v b) = ((a_|_ ^ b_|_) v b)
2		anor3 82	. . . . 5 (a_|_ ^ b_|_) = (a v b)_|_
3	2	ax-r5 37	. . . 4 ((a_|_ ^ b_|_) v b) = ((a v b)_|_ v b)
4	1, 3	ax-r2 35	. . 3 ((a ->5 b) v b) = ((a v b)_|_ v b)
5		oran 79	. . 3 ((a ->5 b) v b) = ((a ->5 b)_|_ ^ b_|_)_|_
6		oran2 84	. . 3 ((a v b)_|_ v b) = ((a v b) ^ b_|_)_|_
7	4, 5, 6	3tr2 61	. 2 ((a ->5 b)_|_ ^ b_|_)_|_ = ((a v b) ^ b_|_)_|_
8	7	con1 63	1 ((a ->5 b)_|_ ^ 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-t 40  df-f 41  df-i5 47  df-le1 122  df-le2 123

metamath.org