Quantifiers and vs if then connective

Quantifiers and vs if then connective

Consider the statement: Jane saw a police officer and Roger saw one too.
The answer given is: $\exists x (P(x) \land S(j,x)) \land\exists x (P(x)
\land S(r,x))$ where P is police, S(w,y) is w saw y. Is this synonymous
with the statement: There exists an x such that if x is a police officer,
then Jane saw the police officer and there exists a y such that if y is a
police officer, then Roger saw the police officer: $\exists x (P(x)
\implies S(j,x)) \land \exists y(P(y)\implies S(r,y))$
It makes sense to me through reasoning but I haven't learned truth tables
for quantifiers and if you rewrite the if then statement into $\neg P \vee
Q$ it seems like it won't work.
What about: Jane saw a police officer, and Roger saw him too. The correct
answer is: $\exists x (P(x) \land S(j,x) \land S(r,x))$ Can I rewrite this
as: There exists an x such that if x is a police officer then Jane saw the
police officer and Roger saw the police officer: $\exists x (P(x) \implies
S(j,x) \land S(r,x))$
Thanks a bunch.