Examples

Let $P \in(A \rightarrow Prop), Q_i \in (B \rightarrow Prop), R \in((A \times B) \rightarrow Prop)$and $C \in Prop.$

1.

$\forall x\!:\!A.\ \forall y\!:\!B.\ (P(x)\ \&\ Q(y)) \Leftrightarrow \forall x\!:\!A.\ P(x)\ \&\ \forall y\!:\!B.\ Q(y)$

2.

$\exists x\!:\!A. P(x)\ \&\ \exists y\!:\!B.\ Q(y) \Rightarrow \exists z\!:\!A \times B.\ R(z)$

3.

$\neg \forall x\!:\!A.\: P(x) \Leftrightarrow \exists x\!:\!A.\: \neg P(x)$

4.

$\forall x\!:\!A.\: (C \Rightarrow P(x)) \Leftrightarrow (C \Rightarrow \forall x\!:\!A.\: P(x))$

5.

$\exists y\!:\!B.\: \forall x\!:\!A.\: R(x, y) \Rightarrow \forall x\!:\!A.\: \exists y\!:\!B.\: R(x, y)$