Question 1207201
Certainly, let's derive the conclusion (∃x)(Ax • Cx) from the given premises using 18 rules of natural deduction and 4 instantiation/generalization rules within 7 steps.

**1. (∃x)(Ax • Bx)** 
   * Premise

**2. (Aa • Ba)** 
   * Existential Instantiation (1)

**3. Ba** 
   * Simplification (2)

**4. Ba ⊃ Ca** 
   * Universal Instantiation (1)

**5. Ca** 
   * Modus Ponens (3, 4)

**6. Aa • Ca** 
   * Conjunction (2, 5)

**7. (∃x)(Ax • Cx)** 
   * Existential Generalization (6)

**Explanation:**

1. **Existential Instantiation (1):** We introduce a new constant 'a' to represent an arbitrary object that satisfies the existential quantifier in premise 1.

2. **Simplification (2):** We extract the conjunct 'Ba' from the conjunction '(Aa • Ba)'.

3. **Universal Instantiation (1):** We instantiate the universal quantifier in premise 1 with the constant 'a'.

4. **Modus Ponens (3, 4):** We apply the rule of Modus Ponens to derive 'Ca' from 'Ba' and 'Ba ⊃ Ca'.

5. **Conjunction (2, 5):** We combine 'Aa' and 'Ca' using the rule of Conjunction.

6. **Existential Generalization (6):** We generalize the statement 'Aa • Ca' to obtain the existential quantifier '(∃x)(Ax • Cx)'.

This derivation demonstrates that the conclusion (∃x)(Ax • Cx) logically follows from the given premises within the specified 7 steps.