Theorem u5lem1n 725

Description: Lemma for unified implication study.

Assertion

Ref	Expression
u5lem1n	((a ->5 b) ->5 a)_|_ = ((a_|_ ^ b) v (a_|_ ^ b_|_))

Proof of Theorem u5lem1n

Step	Hyp	Ref	Expression
1		u5lem1 720	. . 3 ((a ->5 b) ->5 a) = ((a v b) ^ (a v b_|_))
2		ancom 68	. . . 4 ((a v b) ^ (a v b_|_)) = ((a v b_|_) ^ (a v b))
3		df-a 39	. . . . 5 ((a v b_|_) ^ (a v b)) = ((a v b_|_)_|_ v (a v b)_|_)_|_
4		anor2 81	. . . . . . . 8 (a_|_ ^ b) = (a v b_|_)_|_
5		anor3 82	. . . . . . . 8 (a_|_ ^ b_|_) = (a v b)_|_
6	4, 5	2or 67	. . . . . . 7 ((a_|_ ^ b) v (a_|_ ^ b_|_)) = ((a v b_|_)_|_ v (a v b)_|_)
7	6	ax-r4 36	. . . . . 6 ((a_|_ ^ b) v (a_|_ ^ b_|_))_|_ = ((a v b_|_)_|_ v (a v b)_|_)_|_
8	7	ax-r1 34	. . . . 5 ((a v b_|_)_|_ v (a v b)_|_)_|_ = ((a_|_ ^ b) v (a_|_ ^ b_|_))_|_
9	3, 8	ax-r2 35	. . . 4 ((a v b_|_) ^ (a v b)) = ((a_|_ ^ b) v (a_|_ ^ b_|_))_|_
10	2, 9	ax-r2 35	. . 3 ((a v b) ^ (a v b_|_)) = ((a_|_ ^ b) v (a_|_ ^ b_|_))_|_
11	1, 10	ax-r2 35	. 2 ((a ->5 b) ->5 a) = ((a_|_ ^ b) v (a_|_ ^ b_|_))_|_
12	11	con2 64	1 ((a ->5 b) ->5 a)_|_ = ((a_|_ ^ b) v (a_|_ ^ b_|_))

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

This theorem is referenced by:  u5lem2 730

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a4 32  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37  ax-r3 421

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-i5 47  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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