Quantum Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  QLE Home  >  Th. List  >  u4lemc2 GIF version

Theorem u4lemc2 689
 Description: Commutation theorem for non-tollens implication. (Contributed by NM, 14-Dec-1997.)
Hypotheses
Ref Expression
ulemc2.1 a C b
ulemc2.2 a C c
Assertion
Ref Expression
u4lemc2 a C (b4 c)

Proof of Theorem u4lemc2
StepHypRef Expression
1 ulemc2.1 . . . . 5 a C b
2 ulemc2.2 . . . . 5 a C c
31, 2com2an 484 . . . 4 a C (bc)
41comcom2 183 . . . . 5 a C b
54, 2com2an 484 . . . 4 a C (bc)
63, 5com2or 483 . . 3 a C ((bc) ∪ (bc))
74, 2com2or 483 . . . 4 a C (bc)
82comcom2 183 . . . 4 a C c
97, 8com2an 484 . . 3 a C ((bc) ∩ c )
106, 9com2or 483 . 2 a C (((bc) ∪ (bc)) ∪ ((bc) ∩ c ))
11 df-i4 47 . . 3 (b4 c) = (((bc) ∪ (bc)) ∪ ((bc) ∩ c ))
1211ax-r1 35 . 2 (((bc) ∪ (bc)) ∪ ((bc) ∩ c )) = (b4 c)
1310, 12cbtr 182 1 a C (b4 c)
 Colors of variables: term Syntax hints:   C wc 3  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →4 wi4 15 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-i4 47  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by:  u4lemc5  699
 Copyright terms: Public domain W3C validator