Any philosophy majors out there??

Question

I want to know how I can prove the following predicate calculus valid using derivations with UI, HS, Simp....etc

(Ex) (Yx ∙ Zx)

(x)(~Zx v Ax)

:. (Ex) (Ax ∙Yx)

(x)(Vx⊃Wx)

(x)(Wx⊃~Xx)

:. (x)(Xx⊃~Vx)

(x)(Sx⊃Tx)

(X)(Tx⊃Ux)

:. (x)(Sx⊃Ux)

Thanks a lot for your help!!

Answer 1

Here is a proof for the first one. However, you may be using different rules.

1. (∃x)(Yx ∙ Zx)              

2. (x)(~Zx v Ax)....... :. (∃x)(Ax ∙ Yx)

3. Yy ∙ Zy................1, EI

4. Zy ∙ Yy................3, Com

5. Zy.......................4, Simp

6. ~~Zy...................5, DN

7. ~Zy v Ay..............2, UI

8. Ay.......................6,7, DS

9. Yy...................... 3, Simp

10. Ay ∙ Yy..............8,9, Conj

11. (∃x)(Ax ∙ Yx).....10, EG