QLE Home Quantum Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  QLE Home  >  Th. List  >  ud5lem0c GIF version

Theorem ud5lem0c 281
Description: Lemma for unified disjunction. (Contributed by NM, 23-Nov-1997.)
Assertion
Ref Expression
ud5lem0c (a5 b) = (((ab ) ∩ (ab )) ∩ (ab))

Proof of Theorem ud5lem0c
StepHypRef Expression
1 df-i5 48 . . 3 (a5 b) = (((ab) ∪ (ab)) ∪ (ab ))
2 oran 87 . . . 4 (((ab) ∪ (ab)) ∪ (ab )) = (((ab) ∪ (ab)) ∩ (ab ) )
3 oran 87 . . . . . . . 8 ((ab) ∪ (ab)) = ((ab) ∩ (ab) )
4 df-a 40 . . . . . . . . . . 11 (ab) = (ab )
54con2 67 . . . . . . . . . 10 (ab) = (ab )
6 anor2 89 . . . . . . . . . . 11 (ab) = (ab )
76con2 67 . . . . . . . . . 10 (ab) = (ab )
85, 72an 79 . . . . . . . . 9 ((ab) ∩ (ab) ) = ((ab ) ∩ (ab ))
98ax-r4 37 . . . . . . . 8 ((ab) ∩ (ab) ) = ((ab ) ∩ (ab ))
103, 9ax-r2 36 . . . . . . 7 ((ab) ∪ (ab)) = ((ab ) ∩ (ab ))
1110con2 67 . . . . . 6 ((ab) ∪ (ab)) = ((ab ) ∩ (ab ))
12 oran 87 . . . . . . 7 (ab) = (ab )
1312ax-r1 35 . . . . . 6 (ab ) = (ab)
1411, 132an 79 . . . . 5 (((ab) ∪ (ab)) ∩ (ab ) ) = (((ab ) ∩ (ab )) ∩ (ab))
1514ax-r4 37 . . . 4 (((ab) ∪ (ab)) ∩ (ab ) ) = (((ab ) ∩ (ab )) ∩ (ab))
162, 15ax-r2 36 . . 3 (((ab) ∪ (ab)) ∪ (ab )) = (((ab ) ∩ (ab )) ∩ (ab))
171, 16ax-r2 36 . 2 (a5 b) = (((ab ) ∩ (ab )) ∩ (ab))
1817con2 67 1 (a5 b) = (((ab ) ∩ (ab )) ∩ (ab))
Colors of variables: term
Syntax hints:   = wb 1   wn 4  wo 6  wa 7  5 wi5 16
This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38
This theorem depends on definitions:  df-a 40  df-i5 48
This theorem is referenced by:  ud5lem1b  587  ud5lem1c  588  ud5lem3b  592  ud5lem3c  593
  Copyright terms: Public domain W3C validator