Quantum Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  QLE Home  >  Th. List  >  u4lemnoa GIF version

Theorem u4lemnoa 663
 Description: Lemma for non-tollens implication study.
Assertion
Ref Expression
u4lemnoa ((a4 b)a) = ((ab) ∩ (ab ))

Proof of Theorem u4lemnoa
StepHypRef Expression
1 u4lemana 608 . . . 4 ((a4 b) ∩ a ) = ((ab) ∪ (ab ))
2 ax-a2 31 . . . . 5 ((ab) ∪ (ab )) = ((ab ) ∪ (ab))
3 anor3 90 . . . . . 6 (ab ) = (ab)
4 anor2 89 . . . . . 6 (ab) = (ab )
53, 42or 72 . . . . 5 ((ab ) ∪ (ab)) = ((ab) ∪ (ab ) )
62, 5ax-r2 36 . . . 4 ((ab) ∪ (ab )) = ((ab) ∪ (ab ) )
71, 6ax-r2 36 . . 3 ((a4 b) ∩ a ) = ((ab) ∪ (ab ) )
8 anor1 88 . . 3 ((a4 b) ∩ a ) = ((a4 b)a)
9 oran3 93 . . 3 ((ab) ∪ (ab ) ) = ((ab) ∩ (ab ))
107, 8, 93tr2 64 . 2 ((a4 b)a) = ((ab) ∩ (ab ))
1110con1 66 1 ((a4 b)a) = ((ab) ∩ (ab ))
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →4 wi4 15 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-i4 47  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by:  u4lem1  737
 Copyright terms: Public domain W3C validator