When do I use implies vs AND in logic statements? For example,

There is at least one student who has graduated but has not passed at least two courses.

Where $G(x)$ means x has graduated 
and  $P(x, y)$ means x has passed class y
The correct answer is $$∃x∃y∃z\, ((z ≠ y) → (G(x)) ∧ ¬(P(x, y) ∧ ¬(P(x, z)))$$
And I don't see why
$$∃x∃y∃z ((z ≠ y) ∧  (G(x)) ∧ ¬(P(x, y) ∧ ¬(P(x, z)))$$ 
Would be wrong.
Can someone explain the difference?
 A: The "correct" example is really a strange way to think about it. I like to try to make sure that even when I write something in symbolic notation that it makes sense when translated to English. 
What you've got written down as the correct answer sounds something like this in my head: 

There exists x, a student, and two courses y and z such that if y and z aren't equal then the student has failed both y and z and the student has graduated

This is truly a strange way to say that somebody has failed two classes. It's especially strange because of the implication component. We've already picked out the courses when we say there exist some, that's when we make a choice and say "look, this combination of student and two courses exists that fits these criterion" and then we list those criterion. 
What really makes sense in English is something more like 

there's a student who has graduated, and also failed two distinct courses. 

When we use the symbolic notation, we like to introduce the elements up front, so I'll reword it like so: 

there's a student and two classes out there where the student has graduated, the classes are distinct, and the student has failed both classes. 

This quickly translates to what you've got written down. 
Furthermore, what's really wrong with the "correct" statement is that it doesn't actually say there exists a distinct two courses, it just says there exists a student and two courses and IF they're distinct courses then ... 
You see how we haven't actually said two distinct courses exist?
A piece of advise:
Saying what you mean to in words, and gaining clarity, is much more worthwhile than trying to say what you mean in symbols and losing some of your intention. If the symbology doesn't make it more precise or convey more meaning, it's not helping. Make symbolic math work for you by making it convey what you mean in your head and making your statements more clear. 
