Theorem u3lemnoa 644

Description: Lemma for Kalmbach implication study.

Assertion

Ref	Expression
u3lemnoa	((a ->3 b)_|_ v a) = ((a v b) ^ (a v b_|_))

Proof of Theorem u3lemnoa

Step	Hyp	Ref	Expression
1		u3lemana 589	. . . 4 ((a ->3 b) ^ a_|_) = ((a_|_ ^ b) v (a_|_ ^ b_|_))
2		ax-a2 30	. . . . 5 ((a_|_ ^ b) v (a_|_ ^ b_|_)) = ((a_|_ ^ b_|_) v (a_|_ ^ b))
3		anor3 82	. . . . . 6 (a_|_ ^ b_|_) = (a v b)_|_
4		anor2 81	. . . . . 6 (a_|_ ^ b) = (a v b_|_)_|_
5	3, 4	2or 67	. . . . 5 ((a_|_ ^ b_|_) v (a_|_ ^ b)) = ((a v b)_|_ v (a v b_|_)_|_)
6	2, 5	ax-r2 35	. . . 4 ((a_|_ ^ b) v (a_|_ ^ b_|_)) = ((a v b)_|_ v (a v b_|_)_|_)
7	1, 6	ax-r2 35	. . 3 ((a ->3 b) ^ a_|_) = ((a v b)_|_ v (a v b_|_)_|_)
8		anor1 80	. . 3 ((a ->3 b) ^ a_|_) = ((a ->3 b)_|_ v a)_|_
9		oran3 85	. . 3 ((a v b)_|_ v (a v b_|_)_|_) = ((a v b) ^ (a v b_|_))_|_
10	7, 8, 9	3tr2 61	. 2 ((a ->3 b)_|_ v a)_|_ = ((a v b) ^ (a v b_|_))_|_
11	10	con1 63	1 ((a ->3 b)_|_ v a) = ((a v b) ^ (a v b_|_))

Syntax hints:   = wb 1  _|_wn 4   v wo 6   ^ wa 7   ->3 wi3 15

This theorem is referenced by:  u3lem1 718

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-i3 45  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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