QLE Home Quantum Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  QLE Home  >  Th. List  >  u5lemc1 Unicode version
Theorem u5lemc1 684
Description: Commutation theorem for relevance implication.
Assertion
Ref Expression
u5lemc1 a C (a ->5 b)
Proof of Theorem u5lemc1
StepHypRef Expression
1 comanr1 464 . . . 4 a C (a ^ b)
2 comanr1 464 . . . . 5 a' C (a' ^ b)
32comcom6 459 . . . 4 a C (a' ^ b)
41, 3com2or 483 . . 3 a C ((a ^ b) v (a' ^ b))
5 comanr1 464 . . . 4 a' C (a' ^ b')
65comcom6 459 . . 3 a C (a' ^ b')
74, 6com2or 483 . 2 a C (((a ^ b) v (a' ^ b)) v (a' ^ b'))
8 df-i5 48 . . 3 (a ->5 b) = (((a ^ b) v (a' ^ b)) v (a' ^ b'))
98ax-r1 35 . 2 (((a ^ b) v (a' ^ b)) v (a' ^ b')) = (a ->5 b)
107, 9cbtr 182 1 a C (a ->5 b)
Colors of variables: term
Syntax hints:   C wc 3  'wn 4   v wo 6   ^ wa 7   ->5 wi5 16
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-i5 48  df-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by:  u5lemc5  700  u5lembi  725  u5lem1  738  u5lem4  760  u5lem5  765  u5lem6  769
  Copyright terms: Public domain W3C validator