# Difficulty understanding a logical equivalence

1. Feb 22, 2013

### 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?

2. Feb 22, 2013

### Staff: Mentor

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.

3. Feb 22, 2013

### zelmac

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

4. Feb 22, 2013

### Staff: Mentor

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