How can I transform left side into right side in Logic Propositional? I have this formulas:
$$(P \to Q) \to (P \land Q) = (\lnot P \to  Q) \land (Q \to  P), $$
How can I transform left side into right side, or virse-versa, using Logic Propositional?
 A: Use truth table for both logical expressions $(P\to Q)\to (P\wedge Q)$ and $(\lnot P \to  Q) \land (Q \to  P)$.
$$
\begin{array}{|c|c| c|c| c|}
\hline
 P 
   & Q                   
       & (P\to Q)
          &  (P\wedge Q)
            & (P\to Q)\to (P\wedge Q)  
\\\hline
V & V & V & V & V                        
\\\hline
V & F & F & F & V                         
\\\hline
F  & V & V & F & F                         
\\\hline
F & F & V  & F & F                        
\\\hline
\end{array}
$$
and
$$
\begin{array}{|c|c| c|c| c|c| c|}
\hline
 P 
   & Q
     & \lnot P 
       & Q                 
          & (\lnot P \to  Q)
            &  (Q \to  P) 
              & (\lnot P \to  Q) \land (Q \to  P) 
\\\hline
V & V & F & V & V & V & V
\\\hline 
V & F & F & V & V & V & V
\\\hline 
F & V & V & V & V & F & F
\\\hline 
F & F & V & F & F & V & F
\\\hline                      
\end{array}
$$
A: Hint:  Use implication equivalence and distribution to meet in the middle.$$\begin{align} &(P\to Q)\to (P\wedge Q)
\\=~& 
\\=~& 
\\=~&  
\\=~& P\wedge\top & \top\text{ is $``$true"}
\\=~& P 
\\=~& P\vee \bot& \bot\text{ is $``$false"}
\\=~& 
\\=~& 
\\=~& (\neg P\to Q)\wedge(Q\to P)\end{align}$$
