Difficulty understanding a logical equivalence

[zelmac]

$\exists{x}(P(x)\Rightarrow{Q(x)})\equiv{\forall{x}P(x)\Rightarrow{\exists{x}Q(x)}}$

I am able to derive this equivalence by using the standard equivalences of symbolic logic, but when I try to verify this semantically, with an example, I just can't see why these two expressions are equivalent.

Example:
Lets say that x represents days, P(x) represents it's cold on day x, and Q(x) represents it is snowing on day x. If it is true that it is cold every day, and it is true that there is a snowy day, why must it be true that there is a day when, if it is cold, than snow MUST fall?

 

[mfb]

If it is cold every day, and there is at least one snowy day, this snowy day has to be cold. Therefore, there is a day (∃) where it is cold and snow falls.

 

[zelmac]

You concluded:
$\exists{x}(P(x)\wedge{Q(x)})$
and I'm wondering how do you get to:
$\exists{x}(P(x)\Rightarrow{Q(x)})$

 

[mfb]

If P(x) ∧ Q(x) is true for this x, then P(x) => Q(x).

 

[blue_raver22]

I understand your confusion with this logical equivalence. It can be difficult to grasp at first, but let me try to explain it in a different way.

Think of it like this: if P(x) implies Q(x) for all values of x, then it must also be true that if P(x) is true for all values of x, then Q(x) must be true for at least one value of x.

In your example, if it is cold every day (P(x) is true for all values of x), then it must also be true that there is at least one day where if it is cold, then it must snow (Q(x) is true for at least one value of x).

This can be seen as a chain of implications. If P(x) implies Q(x) for all values of x, then we can say that for any specific value of x, if P(x) is true, then Q(x) must also be true. And if P(x) is true for all values of x, then this must hold true for at least one value of x.

I hope this helps clarify the logical equivalence for you. It can be challenging to understand at first, but with practice and examples, it will become clearer.