QLE Home Quantum Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  QLE Home  >  Th. List  >  u2lem8 Unicode version
Theorem u2lem8 782
Description: Lemma for unified implication study.
Assertion
Ref Expression
u2lem8 (a' ->2 (a ->2 (a' ->2 b))) = (a ->2 (a' ->2 b))
Proof of Theorem u2lem8
StepHypRef Expression
1 df-i2 45 . 2 (a' ->2 (a ->2 (a' ->2 b))) = ((a ->2 (a' ->2 b)) v (a'' ^ (a ->2 (a' ->2 b))'))
2 u2lem7 773 . . . 4 (a ->2 (a' ->2 b)) = (((a ^ b') v (a' ^ b')) v b)
3 ax-a1 30 . . . . . . 7 a = a''
43ax-r1 35 . . . . . 6 a'' = a
5 u2lem7n 775 . . . . . 6 (a ->2 (a' ->2 b))' = (((a v b) ^ (a' v b)) ^ b')
64, 52an 79 . . . . 5 (a'' ^ (a ->2 (a' ->2 b))') = (a ^ (((a v b) ^ (a' v b)) ^ b'))
7 an12 81 . . . . . 6 (a ^ (((a v b) ^ (a' v b)) ^ b')) = (((a v b) ^ (a' v b)) ^ (a ^ b'))
8 anass 76 . . . . . . 7 (((a v b) ^ (a' v b)) ^ (a ^ b')) = ((a v b) ^ ((a' v b) ^ (a ^ b')))
9 anor1 88 . . . . . . . . . . 11 (a ^ b') = (a' v b)'
109lan 77 . . . . . . . . . 10 ((a' v b) ^ (a ^ b')) = ((a' v b) ^ (a' v b)')
11 dff 101 . . . . . . . . . . 11 0 = ((a' v b) ^ (a' v b)')
1211ax-r1 35 . . . . . . . . . 10 ((a' v b) ^ (a' v b)') = 0
1310, 12ax-r2 36 . . . . . . . . 9 ((a' v b) ^ (a ^ b')) = 0
1413lan 77 . . . . . . . 8 ((a v b) ^ ((a' v b) ^ (a ^ b'))) = ((a v b) ^ 0)
15 an0 108 . . . . . . . 8 ((a v b) ^ 0) = 0
1614, 15ax-r2 36 . . . . . . 7 ((a v b) ^ ((a' v b) ^ (a ^ b'))) = 0
178, 16ax-r2 36 . . . . . 6 (((a v b) ^ (a' v b)) ^ (a ^ b')) = 0
187, 17ax-r2 36 . . . . 5 (a ^ (((a v b) ^ (a' v b)) ^ b')) = 0
196, 18ax-r2 36 . . . 4 (a'' ^ (a ->2 (a' ->2 b))') = 0
202, 192or 72 . . 3 ((a ->2 (a' ->2 b)) v (a'' ^ (a ->2 (a' ->2 b))')) = ((((a ^ b') v (a' ^ b')) v b) v 0)
21 or0 102 . . . 4 ((((a ^ b') v (a' ^ b')) v b) v 0) = (((a ^ b') v (a' ^ b')) v b)
222ax-r1 35 . . . 4 (((a ^ b') v (a' ^ b')) v b) = (a ->2 (a' ->2 b))
2321, 22ax-r2 36 . . 3 ((((a ^ b') v (a' ^ b')) v b) v 0) = (a ->2 (a' ->2 b))
2420, 23ax-r2 36 . 2 ((a ->2 (a' ->2 b)) v (a'' ^ (a ->2 (a' ->2 b))')) = (a ->2 (a' ->2 b))
251, 24ax-r2 36 1 (a' ->2 (a ->2 (a' ->2 b))) = (a ->2 (a' ->2 b))
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7  0wf 9   ->2 wi2 13
This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38
This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-i2 45  df-le1 130  df-le2 131
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator