We have always problem in formalizing natural language statements.
The first step is how to translate : whenever.
We assume that it has the same meaning of "when".
Thus, the statement is of the form :
When $A$, then $B$
and we symbolize it with the connective : $\rightarrow$ ("if ..., then _") :
$A \rightarrow B$.
Now we need quantifiers for analyzing the two clauses :
The first one will be :
$\exists x(Alert(x) \land Active(x))$
while for the second we have :
$\forall y((Message(y) \land Queued(y)) \rightarrow Transmitted(y))$.
Putting all together :
$$\exists x(Alert(x) \land Active(x)) \rightarrow \forall y((Message(y) \land Queued(y)) \rightarrow Transmitted(y))$$
here are some basic distributive properties in quantifiers, hope it might help someone.
∀x(P(x) ∧ Q(x)) ≡ (∀xP(x) ∧ ∀xQ(x))
∃x(P(x) ∧ Q(x)) → (∃xP(x) ∧ ∃xQ(x))
∀x(P(x) ∨ Q(x)) ← (∀xP(x) ∨ ∀xQ(x))
∃x(P(x) ∨ Q(x)) ≡ (∃xP(x) ∨ ∃xQ(x))
∀x(P(x) → Q(x)) ← (∃xP(x) → ∀xQ(x))
∃x(P(x) → Q(x)) ≡ (∀xP(x) → ∃xQ(x))
∀x¬P(x) ≡ ¬∃xP(x)
∃x¬P(x) ≡ ¬∀xP(x)
∀x∃yT(x,y) ← ∃y∀xT(x,y)
∀x∀yT(x,y) ≡ ∀y∀xT(x,y)
∃x∃yT(x,y) ≡ ∃y∃xT(x,y)
∀x(P(x) ∨ R) ≡ (∀xP(x) ∨ R)
∃x(P(x) ∧ R) ≡ (∃xP(x) ∧ R)
∀x(P(x) → R) ≡ (∃xP(x) → R)
∃x(P(x) → R) → (∀xP(x) → R)
∀x(R → Q(x)) ≡ (R → ∀xQ(x))
∃x(R → Q(x)) → (R → ∃xQ(x))
∀xR ← R
∃xR → R
The following formulas are not valid.
A B counterexample
∃x(P(x) ∧ Q(x)) ← (∃xP(x) ∧ ∃xQ(x)) D = {a, b}, M = {P(a), Q(b)}
∀x(P(x) ∨ Q(x)) → (∀xP(x) ∨ ∀xQ(x)) D = {a, b}, M = {P(a), Q(b)}
∀x(P(x) → Q(x)) → (∃xP(x) → ∀xQ(x)) D = {a, b}, M = {P(a), Q(a)}
∀x∃yT(x,y) → ∃y∀xT(x,y) D = {a, b}, M = {T(a,b), T(b,a)}
∃x(P(x) → R) ← (∀xP(x) → R) D = Ø, M = {R}
∃x(R → Q(x)) ← (R → ∃xQ(x)) D = Ø, M = Ø
∀xR → R D = Ø, M = Ø
∃xR ← R D = Ø, M = {R}
Note: if empty domains are not allowed, then the last four implications are in fact valid.
Best Answer
What he is likely referring to is sometimes you will have relations appear informally inside the quantifier, and those are the rules for how it is translated. For instance,
$(\forall x \in I)P(x)$ informally expresses $(\forall x)(x\in I \to P(x))$
and $(\exists x \in I)P(x)$ informally expresses $(\exists x)(x \in I \land P(x))$
EDIT-
In response to your comment, if he/she is referring to the informal notation above the correct way would be something like the following:
$(\forall x \in I)(\exists y >0)(P(x,y))$ would be translated
$(\forall x)\Big((x \in I) \to \exists y\big((y > 0) \land P(x,y)\big)\Big)$
So you use both. Absent this, I don't know which convention he could be referring to because both can be correct depending on what the statement itself is trying to express. There are plenty of times when you want weaker statements. For instance, you generally want the weakest possible assumptions that suffice to prove a given theorem. This allows you to use the theorem more broadly/frequently as the conditions that must be met to employ it are easier to satisfy.