Theorem u5lemnaa 626

Description: Lemma for relevance implication study.

Assertion

Ref	Expression
u5lemnaa	((a ->5 b)_|_ ^ a) = (a ^ (a_|_ v b_|_))

Proof of Theorem u5lemnaa

Step	Hyp	Ref	Expression
1		anor2 81	. 2 ((a ->5 b)_|_ ^ a) = ((a ->5 b) v a_|_)_|_
2		u5lemona 611	. . . 4 ((a ->5 b) v a_|_) = (a_|_ v (a ^ b))
3	2	ax-r4 36	. . 3 ((a ->5 b) v a_|_)_|_ = (a_|_ v (a ^ b))_|_
4		anor1 80	. . . . 5 (a ^ (a ^ b)_|_) = (a_|_ v (a ^ b))_|_
5	4	ax-r1 34	. . . 4 (a_|_ v (a ^ b))_|_ = (a ^ (a ^ b)_|_)
6		df-a 39	. . . . . 6 (a ^ b) = (a_|_ v b_|_)_|_
7	6	con2 64	. . . . 5 (a ^ b)_|_ = (a_|_ v b_|_)
8	7	lan 70	. . . 4 (a ^ (a ^ b)_|_) = (a ^ (a_|_ v b_|_))
9	5, 8	ax-r2 35	. . 3 (a_|_ v (a ^ b))_|_ = (a ^ (a_|_ v b_|_))
10	3, 9	ax-r2 35	. 2 ((a ->5 b) v a_|_)_|_ = (a ^ (a_|_ v b_|_))
11	1, 10	ax-r2 35	1 ((a ->5 b)_|_ ^ a) = (a ^ (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-t 40  df-f 41  df-i5 47  df-le1 122  df-le2 123

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