Quantum Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  QLE Home  >  Th. List  >  df2i3 GIF version

Theorem df2i3 498
 Description: Alternate definition for Kalmbach implication.
Assertion
Ref Expression
df2i3 (a3 b) = ((ab ) ∪ ((ab) ∩ (a ∪ (ab))))

Proof of Theorem df2i3
StepHypRef Expression
1 df-i3 46 . 2 (a3 b) = (((ab) ∪ (ab )) ∪ (a ∩ (ab)))
2 ax-a3 32 . . 3 (((ab) ∪ (ab )) ∪ (a ∩ (ab))) = ((ab) ∪ ((ab ) ∪ (a ∩ (ab))))
3 or12 80 . . . 4 ((ab) ∪ ((ab ) ∪ (a ∩ (ab)))) = ((ab ) ∪ ((ab) ∪ (a ∩ (ab))))
4 coman1 185 . . . . . . . . . 10 (ab) C a
54comcom 453 . . . . . . . . 9 a C (ab)
65comcom2 183 . . . . . . . 8 a C (ab)
76comcom5 458 . . . . . . 7 a C (ab)
8 comorr 184 . . . . . . . . 9 a C (ab)
98comcom2 183 . . . . . . . 8 a C (ab)
109comcom5 458 . . . . . . 7 a C (ab)
117, 10fh4 472 . . . . . 6 ((ab) ∪ (a ∩ (ab))) = (((ab) ∪ a) ∩ ((ab) ∪ (ab)))
12 lea 160 . . . . . . . . . 10 (ab) ≤ a
13 leo 158 . . . . . . . . . 10 a ≤ (ab)
1412, 13letr 137 . . . . . . . . 9 (ab) ≤ (ab)
1514df-le2 131 . . . . . . . 8 ((ab) ∪ (ab)) = (ab)
1615lan 77 . . . . . . 7 (((ab) ∪ a) ∩ ((ab) ∪ (ab))) = (((ab) ∪ a) ∩ (ab))
17 ancom 74 . . . . . . . 8 (((ab) ∪ a) ∩ (ab)) = ((ab) ∩ ((ab) ∪ a))
18 ax-a2 31 . . . . . . . . 9 ((ab) ∪ a) = (a ∪ (ab))
1918lan 77 . . . . . . . 8 ((ab) ∩ ((ab) ∪ a)) = ((ab) ∩ (a ∪ (ab)))
2017, 19ax-r2 36 . . . . . . 7 (((ab) ∪ a) ∩ (ab)) = ((ab) ∩ (a ∪ (ab)))
2116, 20ax-r2 36 . . . . . 6 (((ab) ∪ a) ∩ ((ab) ∪ (ab))) = ((ab) ∩ (a ∪ (ab)))
2211, 21ax-r2 36 . . . . 5 ((ab) ∪ (a ∩ (ab))) = ((ab) ∩ (a ∪ (ab)))
2322lor 70 . . . 4 ((ab ) ∪ ((ab) ∪ (a ∩ (ab)))) = ((ab ) ∪ ((ab) ∩ (a ∪ (ab))))
243, 23ax-r2 36 . . 3 ((ab) ∪ ((ab ) ∪ (a ∩ (ab)))) = ((ab ) ∪ ((ab) ∩ (a ∪ (ab))))
252, 24ax-r2 36 . 2 (((ab) ∪ (ab )) ∪ (a ∩ (ab))) = ((ab ) ∪ ((ab) ∩ (a ∪ (ab))))
261, 25ax-r2 36 1 (a3 b) = ((ab ) ∪ ((ab) ∩ (a ∪ (ab))))
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ 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:  i3n2  501  ni32  502  i3lem1  504  i3th1  543  i3orlem5  556
 Copyright terms: Public domain W3C validator