Quantum Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  QLE Home  >  Th. List  >  u2lemanb GIF version

Theorem u2lemanb 616
 Description: Lemma for Dishkant implication study. (Contributed by NM, 14-Dec-1997.)
Assertion
Ref Expression
u2lemanb ((a2 b) ∩ b ) = (ab )

Proof of Theorem u2lemanb
StepHypRef Expression
1 df-i2 45 . . 3 (a2 b) = (b ∪ (ab ))
21ran 78 . 2 ((a2 b) ∩ b ) = ((b ∪ (ab )) ∩ b )
3 comid 187 . . . . 5 b C b
43comcom3 454 . . . 4 b C b
5 comanr2 465 . . . 4 b C (ab )
64, 5fh1r 473 . . 3 ((b ∪ (ab )) ∩ b ) = ((bb ) ∪ ((ab ) ∩ b ))
7 ax-a2 31 . . . 4 ((bb ) ∪ ((ab ) ∩ b )) = (((ab ) ∩ b ) ∪ (bb ))
8 anass 76 . . . . . . 7 ((ab ) ∩ b ) = (a ∩ (bb ))
9 anidm 111 . . . . . . . 8 (bb ) = b
109lan 77 . . . . . . 7 (a ∩ (bb )) = (ab )
118, 10ax-r2 36 . . . . . 6 ((ab ) ∩ b ) = (ab )
12 dff 101 . . . . . . 7 0 = (bb )
1312ax-r1 35 . . . . . 6 (bb ) = 0
1411, 132or 72 . . . . 5 (((ab ) ∩ b ) ∪ (bb )) = ((ab ) ∪ 0)
15 or0 102 . . . . 5 ((ab ) ∪ 0) = (ab )
1614, 15ax-r2 36 . . . 4 (((ab ) ∩ b ) ∪ (bb )) = (ab )
177, 16ax-r2 36 . . 3 ((bb ) ∪ ((ab ) ∩ b )) = (ab )
186, 17ax-r2 36 . 2 ((b ∪ (ab )) ∩ b ) = (ab )
192, 18ax-r2 36 1 ((a2 b) ∩ b ) = (ab )
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7  0wf 9   →2 wi2 13 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-i2 45  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by:  u2lemnob  671  u21lembi  727  bi3  839  bi4  840  imp3  841  oal42  935  oa23  936
 Copyright terms: Public domain W3C validator