Show that $[(P→Q)∧P)]→Q$ is a tautology using rules of replacement

Question

Show that $[(P→Q)∧P)]→Q$ is a tautology using rules of replacement

I've done this so far,

• from $[(P→Q)∧P)]→Q$ to $[(~P∧Q)∧P)]→Q$ by Mat. Imp.
• to $[P∧(~P∧Q))]→Q$ by Commutation.

After that I'm thinking of using distribution although I'm not so sure what the result will be.

Answer 1

One approach: \begin{align} [[(P \Rightarrow Q) \land P] \Rightarrow Q] &\Leftrightarrow [[(\lnot P \lor Q) \land P] \Rightarrow Q] \\ &\Leftrightarrow [[(\lnot P \lor Q) \land (\lnot (\lnot P))] \Rightarrow Q] \\ &\Leftrightarrow [Q \Rightarrow Q] \end{align}

which is an easily verifiable tautology.