QLE Home Quantum Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  QLE Home  >  Th. List  >  u4lemonb Unicode version
Theorem u4lemonb 638
Description: Lemma for non-tollens implication study.
Assertion
Ref Expression
u4lemonb ((a ->4 b) v b') = (((a ^ b) v (a' ^ b)) v b')
Proof of Theorem u4lemonb
StepHypRef Expression
1 df-i4 47 . . 3 (a ->4 b) = (((a ^ b) v (a' ^ b)) v ((a' v b) ^ b'))
21ax-r5 38 . 2 ((a ->4 b) v b') = ((((a ^ b) v (a' ^ b)) v ((a' v b) ^ b')) v b')
3 ax-a3 32 . . 3 ((((a ^ b) v (a' ^ b)) v ((a' v b) ^ b')) v b') = (((a ^ b) v (a' ^ b)) v (((a' v b) ^ b') v b'))
4 lear 161 . . . . 5 ((a' v b) ^ b') =< b'
54df-le2 131 . . . 4 (((a' v b) ^ b') v b') = b'
65lor 70 . . 3 (((a ^ b) v (a' ^ b)) v (((a' v b) ^ b') v b')) = (((a ^ b) v (a' ^ b)) v b')
73, 6ax-r2 36 . 2 ((((a ^ b) v (a' ^ b)) v ((a' v b) ^ b')) v b') = (((a ^ b) v (a' ^ b)) v b')
82, 7ax-r2 36 1 ((a ->4 b) v b') = (((a ^ b) v (a' ^ b)) v b')
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7   ->4 wi4 15
This theorem was proved from axioms:  ax-a2 31  ax-a3 32  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38
This theorem depends on definitions:  df-a 40  df-i4 47  df-le1 130  df-le2 131
This theorem is referenced by:  u4lemnab  653  u4lem3  752
  Copyright terms: Public domain W3C validator