A slightly different approach: you need to show you cannot make an assignment of truth values to the component/atomic sentences that simultaneously make your premises true and your conclusions false. Then consider all assignments of component sentences that falsify your conclusion ( if there is no such assignment, your statement is a tautology), then check that any of these assignments to the premise sentences cannot return you a truth value T:
So we need to falsify:
So we need the assignments:
i)q:=F and (¬p∧r):=F , so one of the two is false, and r is true, so we have:
Are the only assignments that falsify the conclusion. We now need to check that this assignment on the antecedent does not give a truth value T
So we check:
i.1) returns false, since q,r are both false, then (q∨r) is false, and the conjunction of the three is false.
We also get a false, because if p is true and q is false, then p->q is false. Again, the conjunction is false, and we conclude the argument is valid, i.e., that there is no assignment of truth values to the component sentences that makes the premises true and the conclusions false.