# Difficulty understanding a logical equivalence

## Main Question or Discussion Point

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

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?

## Answers and Replies

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mfb
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.

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