QLE Home Quantum Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  QLE Home  >  Th. List  >  u3lemoa Unicode version
Theorem u3lemoa 622
Description: Lemma for Kalmbach implication study.
Assertion
Ref Expression
u3lemoa ((a ->3 b) v a) = (a v ((a' ^ b) v (a' ^ b')))
Proof of Theorem u3lemoa
StepHypRef Expression
1 df-i3 46 . . 3 (a ->3 b) = (((a' ^ b) v (a' ^ b')) v (a ^ (a' v b)))
21ax-r5 38 . 2 ((a ->3 b) v a) = ((((a' ^ b) v (a' ^ b')) v (a ^ (a' v b))) v a)
3 ax-a3 32 . . 3 ((((a' ^ b) v (a' ^ b')) v (a ^ (a' v b))) v a) = (((a' ^ b) v (a' ^ b')) v ((a ^ (a' v b)) v a))
4 lea 160 . . . . . 6 (a ^ (a' v b)) =< a
54df-le2 131 . . . . 5 ((a ^ (a' v b)) v a) = a
65lor 70 . . . 4 (((a' ^ b) v (a' ^ b')) v ((a ^ (a' v b)) v a)) = (((a' ^ b) v (a' ^ b')) v a)
7 ax-a2 31 . . . 4 (((a' ^ b) v (a' ^ b')) v a) = (a v ((a' ^ b) v (a' ^ b')))
86, 7ax-r2 36 . . 3 (((a' ^ b) v (a' ^ b')) v ((a ^ (a' v b)) v a)) = (a v ((a' ^ b) v (a' ^ b')))
93, 8ax-r2 36 . 2 ((((a' ^ b) v (a' ^ b')) v (a ^ (a' v b))) v a) = (a v ((a' ^ b) v (a' ^ b')))
102, 9ax-r2 36 1 ((a ->3 b) v a) = (a v ((a' ^ b) v (a' ^ b')))
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7   ->3 wi3 14
This theorem was proved from axioms:  ax-a2 31  ax-a3 32  ax-a5 34  ax-r1 35  ax-r2 36  ax-r5 38
This theorem depends on definitions:  df-a 40  df-i3 46  df-le1 130  df-le2 131
This theorem is referenced by:  u3lemnana  647
  Copyright terms: Public domain W3C validator