My teacher told me these things :

There exist x belongs to R , ( p(x) ^ q(x) ) iff ( there exist x belongs R , p(x) ) ^ ( there exist x belongs R , q(x) ) is not logically correct.

But

There exist x belongs to R ( p(x) V q(x) ) iff ( there exist x belongs R , p(x) ) V ( there exist x belongs R , q(x) ) is right.

For my knowledge I guess in the first statement,

There exist x belongs to R , ( p(x) ^ q(x) ) implies ( there exist x belongs R , p(x) ) ^ ( there exist x belongs R , q(x) ) is logically correct.

But the other way that means , ( there exist x belongs R , p(x) ) ^ ( there exist x belongs R , q(x) ) implies There exist x belongs to R , ( p(x) ^ q(x) ) is wrong. Am I right with this?

In the second one both the implies parts are correct.

But is there a standard method of proving such things? If so can someone please explain.