Discrete Mathematics Proofs for ∃ and ∀

Question

Premises or Givens:

• $∃x(A(x) → B(x))$
• $∀x (B(x) → K(x))$

To Prove:

• $∃x(A(x) → K(x))$

My Solution:

1. $A(z) → B(z)$ From premise and Existential instantiation $x$ for $z$

2. $B(z) → K(z)$ From premise and Universal instantiation $x$ for $z$

3. $A(z) → K(z)$ Transitivity of 1,2

4. $∃x(A(x) → K(x))$ From Existential generalization (Substitute $z$ for $x$)

OR

I was thinking about assuming $A(z)$ and then using Modus Ponens to get $B(z)$ and then further $K(z)$, then using the deduction theorem on $A(z)$ and $K(z)$, and then using Existential Generalization on that statement by substituting $x$ for $z$.

Can someone suggest which way would be more effective?

Is there any other effective way to solve it?