Quantum Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  QLE Home  >  Th. List  >  u3lem14mp GIF version

Theorem u3lem14mp 794
 Description: Used to prove →1 modus ponens rule in →3 system.
Assertion
Ref Expression
u3lem14mp ((a3 b )3 (a3 (a3 b))) = 1

Proof of Theorem u3lem14mp
StepHypRef Expression
1 lear 161 . . . 4 (((ab ) ∩ (ab )) ∩ (a ∪ (ab ))) ≤ (a ∪ (ab ))
2 lear 161 . . . . . 6 (ab ) ≤ b
3 ax-a1 30 . . . . . . 7 b = b
43ax-r1 35 . . . . . 6 b = b
52, 4lbtr 139 . . . . 5 (ab ) ≤ b
65lelor 166 . . . 4 (a ∪ (ab )) ≤ (ab)
71, 6letr 137 . . 3 (((ab ) ∩ (ab )) ∩ (a ∪ (ab ))) ≤ (ab)
8 ud3lem0c 279 . . 3 (a3 b ) = (((ab ) ∩ (ab )) ∩ (a ∪ (ab )))
9 u3lem5 763 . . 3 (a3 (a3 b)) = (ab)
107, 8, 9le3tr1 140 . 2 (a3 b ) ≤ (a3 (a3 b))
1110u3lemle1 712 1 ((a3 b )3 (a3 (a3 b))) = 1
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   →3 wi3 14 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-i3 46  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator