Symbolize and provide derivations to show the equivalence for the following pairs of sentences.

1.  Not everyone who studies will pass.  Someone who did not pass had studied..

~(x)(Sx -> Px)   and   (Ex)(~Px & Sx)
1.  ~(x)(Sx -> Px)        \show  (Ex)(~Px & Sx)
2.  (Ex)(Sx & ~Px)      1,  QN
3.  (Ex)(~Px & Sx)      2,  Com

1.  (Ex)(~Px & Sx)      \show ~(x)(Sx -> Px)
2.  ~(x)(~Px -> ~Sx)    1,  QN
3.  ~(x)(Sx -> Px)        2,  Trans


2.  Unless everyone is ambitious there will be individuals who do not succeed.  If everyone succeeds then everyone was ambitious.

(x)Ax v (Ex)~Sx   and   (x)Sx -> (x)Ax
1.  (x)Ax v (Ex)~Sx         \show (x)Sx -> (x)Ax
2.  (Ex)~Sx v  (x)Ax        1,  Com
3.  ~(Ex)~Sx -> (x)Ax      2,  Impl
4.  (x)Sx -> (x)Ax            3,  QN

1.  (x)Sx -> (x)Ax             \show (x)Ax v (Ex)~Sx
2.  ~(x)Sx v (x)Ax             1,  Impl
3.  (Ex)~Sx v (x)Ax           2,  QN
4.  (x)Ax v (Ex)~Sx           3,  Com

Do note that other derivations can be presented in which the steps are taken in a different order.  All that matters is that any step is justified by looking back to a call line earlier in the proof.