QLE Home Quantum Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  QLE Home  >  Th. List  >  i3orlem1 GIF version

Theorem i3orlem1 552
Description: Lemma for Kalmbach implication OR builder. (Contributed by NM, 11-Nov-1997.)
Assertion
Ref Expression
i3orlem1 ((ac) ∩ ((ac) ∪ (bc))) ≤ ((ac) →3 (bc))

Proof of Theorem i3orlem1
StepHypRef Expression
1 leor 159 . 2 ((ac) ∩ ((ac) ∪ (bc))) ≤ ((((ac) ∩ (bc)) ∪ ((ac) ∩ (bc) )) ∪ ((ac) ∩ ((ac) ∪ (bc))))
2 df-i3 46 . . 3 ((ac) →3 (bc)) = ((((ac) ∩ (bc)) ∪ ((ac) ∩ (bc) )) ∪ ((ac) ∩ ((ac) ∪ (bc))))
32ax-r1 35 . 2 ((((ac) ∩ (bc)) ∪ ((ac) ∩ (bc) )) ∪ ((ac) ∩ ((ac) ∪ (bc)))) = ((ac) →3 (bc))
41, 3lbtr 139 1 ((ac) ∩ ((ac) ∪ (bc))) ≤ ((ac) →3 (bc))
Colors of variables: term
Syntax hints:  wle 2   wn 4  wo 6  wa 7  3 wi3 14
This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38
This theorem depends on definitions:  df-a 40  df-i3 46  df-le1 130  df-le2 131
This theorem is referenced by:  i3orlem2  553  i3orlem3  554
  Copyright terms: Public domain W3C validator