Theorem u2lemona 608

Description: Lemma for Dishkant implication study.

Assertion

Ref	Expression
u2lemona	((a ->2 b) v a_|_) = (a_|_ v b)

Proof of Theorem u2lemona

Step	Hyp	Ref	Expression
1		df-i2 44	. . 3 (a ->2 b) = (b v (a_|_ ^ b_|_))
2	1	ax-r5 37	. 2 ((a ->2 b) v a_|_) = ((b v (a_|_ ^ b_|_)) v a_|_)
3		ax-a3 31	. . 3 ((b v (a_|_ ^ b_|_)) v a_|_) = (b v ((a_|_ ^ b_|_) v a_|_))
4		ax-a2 30	. . . 4 (b v ((a_|_ ^ b_|_) v a_|_)) = (((a_|_ ^ b_|_) v a_|_) v b)
5		lea 152	. . . . . 6 (a_|_ ^ b_|_) =< a_|_
6	5	df-le2 123	. . . . 5 ((a_|_ ^ b_|_) v a_|_) = a_|_
7	6	ax-r5 37	. . . 4 (((a_|_ ^ b_|_) v a_|_) v b) = (a_|_ v b)
8	4, 7	ax-r2 35	. . 3 (b v ((a_|_ ^ b_|_) v a_|_)) = (a_|_ v b)
9	3, 8	ax-r2 35	. 2 ((b v (a_|_ ^ b_|_)) v a_|_) = (a_|_ v b)
10	2, 9	ax-r2 35	1 ((a ->2 b) v a_|_) = (a_|_ v b)

Syntax hints:   = wb 1  _|_wn 4   v wo 6   ^ wa 7   ->2 wi2 14

This theorem is referenced by:  u2lemnaa 623

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-i2 44  df-le1 122  df-le2 123

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