Theorem u1lemnaa 622

Description: Lemma for Sasaki implication study.

Assertion

Ref	Expression
u1lemnaa	((a ->1 b)_|_ ^ a) = (a ^ (a_|_ v b_|_))

Proof of Theorem u1lemnaa

Step	Hyp	Ref	Expression
1		anor2 81	. 2 ((a ->1 b)_|_ ^ a) = ((a ->1 b) v a_|_)_|_
2		u1lemona 607	. . . 4 ((a ->1 b) v a_|_) = (a_|_ v (a ^ b))
3	2	ax-r4 36	. . 3 ((a ->1 b) v a_|_)_|_ = (a_|_ v (a ^ b))_|_
4		df-a 39	. . . . 5 (a ^ (a_|_ v b_|_)) = (a_|_ v (a_|_ v b_|_)_|_)_|_
5		df-a 39	. . . . . . . 8 (a ^ b) = (a_|_ v b_|_)_|_
6	5	lor 66	. . . . . . 7 (a_|_ v (a ^ b)) = (a_|_ v (a_|_ v b_|_)_|_)
7	6	ax-r4 36	. . . . . 6 (a_|_ v (a ^ b))_|_ = (a_|_ v (a_|_ v b_|_)_|_)_|_
8	7	ax-r1 34	. . . . 5 (a_|_ v (a_|_ v b_|_)_|_)_|_ = (a_|_ v (a ^ b))_|_
9	4, 8	ax-r2 35	. . . 4 (a ^ (a_|_ v b_|_)) = (a_|_ v (a ^ b))_|_
10	9	ax-r1 34	. . 3 (a_|_ v (a ^ b))_|_ = (a ^ (a_|_ v b_|_))
11	3, 10	ax-r2 35	. 2 ((a ->1 b) v a_|_)_|_ = (a ^ (a_|_ v b_|_))
12	1, 11	ax-r2 35	1 ((a ->1 b)_|_ ^ a) = (a ^ (a_|_ v b_|_))

Syntax hints:   = wb 1  _|_wn 4   v wo 6   ^ wa 7   ->1 wi1 13

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-i1 43

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