QLE Home Quantum Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  QLE Home  >  Th. List  >  u3lem1n Unicode version
Theorem u3lem1n 741
Description: Lemma for unified implication study.
Assertion
Ref Expression
u3lem1n ((a ->3 b) ->3 a)' = ((a' ^ b) v (a' ^ b'))
Proof of Theorem u3lem1n
StepHypRef Expression
1 u3lem1 736 . . 3 ((a ->3 b) ->3 a) = ((a v b) ^ (a v b'))
2 ancom 74 . . . 4 ((a v b) ^ (a v b')) = ((a v b') ^ (a v b))
3 df-a 40 . . . . 5 ((a v b') ^ (a v b)) = ((a v b')' v (a v b)')'
4 anor2 89 . . . . . . . 8 (a' ^ b) = (a v b')'
5 anor3 90 . . . . . . . 8 (a' ^ b') = (a v b)'
64, 52or 72 . . . . . . 7 ((a' ^ b) v (a' ^ b')) = ((a v b')' v (a v b)')
76ax-r4 37 . . . . . 6 ((a' ^ b) v (a' ^ b'))' = ((a v b')' v (a v b)')'
87ax-r1 35 . . . . 5 ((a v b')' v (a v b)')' = ((a' ^ b) v (a' ^ b'))'
93, 8ax-r2 36 . . . 4 ((a v b') ^ (a v b)) = ((a' ^ b) v (a' ^ b'))'
102, 9ax-r2 36 . . 3 ((a v b) ^ (a v b')) = ((a' ^ b) v (a' ^ b'))'
111, 10ax-r2 36 . 2 ((a ->3 b) ->3 a) = ((a' ^ b) v (a' ^ b'))'
1211con2 67 1 ((a ->3 b) ->3 a)' = ((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-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  ax-r3 439
This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i3 46  df-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by:  u3lem2  746
  Copyright terms: Public domain W3C validator