Tricky predicate logic problem

Question

I'm having a hard time proving that this is a valid argument

$Premise 1: (Ǝx)Kx→(\forall x)(Lx→Mx)$

$Premise 2: Kc • Lc$

$Conclusion: Mc$

I am getting confused with all the existential/universal instantiation rules.

First I did: $Kc\to\forall x(Lx\to Mx)$ by existential instantiation

then $Kc\to Lc\to Mc$ by universal instantiation

therefore $Mc$

Is there anywhere where I'm supposed to infer modus ponens since I've already been given Kc and Lc in premise 2??

Answer 1

Hint : assuming you are using Natural Deduction ...

From Premise-2 : $Kc \land Lc$ you have to derive $Kc$ (by $\land$-elimination), followed by $\exists x Kx$ (by $\exists$-introduction).

In this way, you can use $\to$-elimination (i.e. modus ponens) with Premise-1 and derive : $(∀x)(Lx \to Mx)$.

Now you have to use $\forall$-elimination (i.e. universal instantiation) with $c$ to get : $Lc \to Mc$.