Quantum Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  QLE Home  >  Th. List  >  u4lemonb GIF version

Theorem u4lemonb 638
 Description: Lemma for non-tollens implication study. (Contributed by NM, 15-Dec-1997.)
Assertion
Ref Expression
u4lemonb ((a4 b) ∪ b ) = (((ab) ∪ (ab)) ∪ b )

Proof of Theorem u4lemonb
StepHypRef Expression
1 df-i4 47 . . 3 (a4 b) = (((ab) ∪ (ab)) ∪ ((ab) ∩ b ))
21ax-r5 38 . 2 ((a4 b) ∪ b ) = ((((ab) ∪ (ab)) ∪ ((ab) ∩ b )) ∪ b )
3 ax-a3 32 . . 3 ((((ab) ∪ (ab)) ∪ ((ab) ∩ b )) ∪ b ) = (((ab) ∪ (ab)) ∪ (((ab) ∩ b ) ∪ b ))
4 lear 161 . . . . 5 ((ab) ∩ b ) ≤ b
54df-le2 131 . . . 4 (((ab) ∩ b ) ∪ b ) = b
65lor 70 . . 3 (((ab) ∪ (ab)) ∪ (((ab) ∩ b ) ∪ b )) = (((ab) ∪ (ab)) ∪ b )
73, 6ax-r2 36 . 2 ((((ab) ∪ (ab)) ∪ ((ab) ∩ b )) ∪ b ) = (((ab) ∪ (ab)) ∪ b )
82, 7ax-r2 36 1 ((a4 b) ∪ b ) = (((ab) ∪ (ab)) ∪ b )
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →4 wi4 15 This theorem was proved from axioms:  ax-a2 31  ax-a3 32  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38 This theorem depends on definitions:  df-a 40  df-i4 47  df-le1 130  df-le2 131 This theorem is referenced by:  u4lemnab  653  u4lem3  752
 Copyright terms: Public domain W3C validator