For all of the following problems, give proofs using the naturaldeduction system we have learned in this class. Use only the rulesgiven in lecture 4 (&I, &E, vI, vE, →E).

Rule 1: CONDITIONAL ELIMINATION (Modus ponens)

Rule 2: CONJUNCTION ELIMINATION

Rule 3: CONJUNCTION INTRODUCTION

Rule 4: DISJUNCTION INTRODUCTION

Rule 5: DISJUNCTION ELIMINATION

From premises P, (PvQ)→~R, and (~R&P)→(RvQ), prove Q.