Theorem u5lemoa 606

Description: Lemma for relevance implication study.

Assertion

Ref	Expression
u5lemoa	((a ->5 b) v a) = (a v ((a_|_ ^ b) v (a_|_ ^ b_|_)))

Proof of Theorem u5lemoa

Step	Hyp	Ref	Expression
1		df-i5 47	. . 3 (a ->5 b) = (((a ^ b) v (a_|_ ^ b)) v (a_|_ ^ b_|_))
2	1	ax-r5 37	. 2 ((a ->5 b) v a) = ((((a ^ b) v (a_|_ ^ b)) v (a_|_ ^ b_|_)) v a)
3		ax-a2 30	. . 3 ((((a ^ b) v (a_|_ ^ b)) v (a_|_ ^ b_|_)) v a) = (a v (((a ^ b) v (a_|_ ^ b)) v (a_|_ ^ b_|_)))
4		ax-a3 31	. . . . 5 (((a ^ b) v (a_|_ ^ b)) v (a_|_ ^ b_|_)) = ((a ^ b) v ((a_|_ ^ b) v (a_|_ ^ b_|_)))
5	4	lor 66	. . . 4 (a v (((a ^ b) v (a_|_ ^ b)) v (a_|_ ^ b_|_))) = (a v ((a ^ b) v ((a_|_ ^ b) v (a_|_ ^ b_|_))))
6		ax-a3 31	. . . . . 6 ((a v (a ^ b)) v ((a_|_ ^ b) v (a_|_ ^ b_|_))) = (a v ((a ^ b) v ((a_|_ ^ b) v (a_|_ ^ b_|_))))
7	6	ax-r1 34	. . . . 5 (a v ((a ^ b) v ((a_|_ ^ b) v (a_|_ ^ b_|_)))) = ((a v (a ^ b)) v ((a_|_ ^ b) v (a_|_ ^ b_|_)))
8		a5b 112	. . . . . 6 (a v (a ^ b)) = a
9	8	ax-r5 37	. . . . 5 ((a v (a ^ b)) v ((a_|_ ^ b) v (a_|_ ^ b_|_))) = (a v ((a_|_ ^ b) v (a_|_ ^ b_|_)))
10	7, 9	ax-r2 35	. . . 4 (a v ((a ^ b) v ((a_|_ ^ b) v (a_|_ ^ b_|_)))) = (a v ((a_|_ ^ b) v (a_|_ ^ b_|_)))
11	5, 10	ax-r2 35	. . 3 (a v (((a ^ b) v (a_|_ ^ b)) v (a_|_ ^ b_|_))) = (a v ((a_|_ ^ b) v (a_|_ ^ b_|_)))
12	3, 11	ax-r2 35	. 2 ((((a ^ b) v (a_|_ ^ b)) v (a_|_ ^ b_|_)) v a) = (a v ((a_|_ ^ b) v (a_|_ ^ b_|_)))
13	2, 12	ax-r2 35	1 ((a ->5 b) v a) = (a v ((a_|_ ^ b) v (a_|_ ^ b_|_)))

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

This theorem is referenced by:  u5lemnana 631

This theorem was proved from axioms:  ax-a2 30  ax-a3 31  ax-a5 33  ax-r1 34  ax-r2 35  ax-r5 37

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

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