Quantum Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  QLE Home  >  Th. List  >  gomaex3h10 GIF version

Theorem gomaex3h10 911
 Description: Hypothesis for Godowski 6-var -> Mayet Example 3. (Contributed by NM, 29-Nov-1999.)
Hypotheses
Ref Expression
gomaex3h10.10 q = ((ef) →1 (bc) )
gomaex3h10.21 x = q
gomaex3h10.22 y = (ef)
Assertion
Ref Expression
gomaex3h10 xy

Proof of Theorem gomaex3h10
StepHypRef Expression
1 lea 160 . . 3 ((ef) ∩ ((ef) ∩ (bc) ) ) ≤ (ef)
2 gomaex3h10.10 . . . 4 q = ((ef) →1 (bc) )
3 df-i1 44 . . . . . 6 ((ef) →1 (bc) ) = ((ef) ∪ ((ef) ∩ (bc) ))
43ax-r4 37 . . . . 5 ((ef) →1 (bc) ) = ((ef) ∪ ((ef) ∩ (bc) ))
5 anor1 88 . . . . . 6 ((ef) ∩ ((ef) ∩ (bc) ) ) = ((ef) ∪ ((ef) ∩ (bc) ))
65ax-r1 35 . . . . 5 ((ef) ∪ ((ef) ∩ (bc) )) = ((ef) ∩ ((ef) ∩ (bc) ) )
74, 6ax-r2 36 . . . 4 ((ef) →1 (bc) ) = ((ef) ∩ ((ef) ∩ (bc) ) )
82, 7ax-r2 36 . . 3 q = ((ef) ∩ ((ef) ∩ (bc) ) )
9 ax-a1 30 . . . 4 (ef) = (ef)
109ax-r1 35 . . 3 (ef) = (ef)
111, 8, 10le3tr1 140 . 2 q ≤ (ef)
12 gomaex3h10.21 . 2 x = q
13 gomaex3h10.22 . . 3 y = (ef)
1413ax-r4 37 . 2 y = (ef)
1511, 12, 14le3tr1 140 1 xy
 Colors of variables: term Syntax hints:   = wb 1   ≤ wle 2  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →1 wi1 12 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-i1 44  df-le1 130  df-le2 131 This theorem is referenced by:  gomaex3lem5  918
 Copyright terms: Public domain W3C validator