# Proof Using Truth Tables

Pleae forgive the very basic question, but I know nothing really of formal logic and so would appreciate some feedback.

The truth table defining the implication operator

P    Q    P implies Q
T    T         T
T    F         F
F    T         T
F    F         T


together with the negation operator ~ defined in the obvious way enables one to construct the following table for ~Q $\implies$ ~P:

P  Q  ~P  ~Q  ~Q implies ~P
T  T   F   F        T
T  F   F   T        F
F  T   T   F        T
F  F   T   T        T


Evidently, truth values for ~Q $\implies$ ~P are the same as those for P $\implies$ Q. Is this enough to prove that $P \implies Q$ if and only if ~Q $\implies$ ~P ? My thinking is that yes, it is, because I belive that the logical operators involved are defined by their respective truth tables and this being the case the observations above should be sufficient to prove the equivalence.

-