QLE Home Quantum Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  QLE Home  >  Th. List  >  ud3lem3b Unicode version
Theorem ud3lem3b 573
Description: Lemma for unified disjunction.
Assertion
Ref Expression
ud3lem3b ((a ->3 b)' ^ (a v b)') = 0
Proof of Theorem ud3lem3b
StepHypRef Expression
1 ud3lem0c 279 . . 3 (a ->3 b)' = (((a v b') ^ (a v b)) ^ (a' v (a ^ b')))
21ran 78 . 2 ((a ->3 b)' ^ (a v b)') = ((((a v b') ^ (a v b)) ^ (a' v (a ^ b'))) ^ (a v b)')
3 an32 83 . . 3 ((((a v b') ^ (a v b)) ^ (a' v (a ^ b'))) ^ (a v b)') = ((((a v b') ^ (a v b)) ^ (a v b)') ^ (a' v (a ^ b')))
4 anass 76 . . . . . 6 (((a v b') ^ (a v b)) ^ (a v b)') = ((a v b') ^ ((a v b) ^ (a v b)'))
5 dff 101 . . . . . . . . 9 0 = ((a v b) ^ (a v b)')
65ax-r1 35 . . . . . . . 8 ((a v b) ^ (a v b)') = 0
76lan 77 . . . . . . 7 ((a v b') ^ ((a v b) ^ (a v b)')) = ((a v b') ^ 0)
8 an0 108 . . . . . . 7 ((a v b') ^ 0) = 0
97, 8ax-r2 36 . . . . . 6 ((a v b') ^ ((a v b) ^ (a v b)')) = 0
104, 9ax-r2 36 . . . . 5 (((a v b') ^ (a v b)) ^ (a v b)') = 0
1110ran 78 . . . 4 ((((a v b') ^ (a v b)) ^ (a v b)') ^ (a' v (a ^ b'))) = (0 ^ (a' v (a ^ b')))
12 an0r 109 . . . 4 (0 ^ (a' v (a ^ b'))) = 0
1311, 12ax-r2 36 . . 3 ((((a v b') ^ (a v b)) ^ (a v b)') ^ (a' v (a ^ b'))) = 0
143, 13ax-r2 36 . 2 ((((a v b') ^ (a v b)) ^ (a' v (a ^ b'))) ^ (a v b)') = 0
152, 14ax-r2 36 1 ((a ->3 b)' ^ (a v b)') = 0
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7  0wf 9   ->3 wi3 14
This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  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-i3 46
This theorem is referenced by:  ud3lem3  576
  Copyright terms: Public domain W3C validator