$$\forall d\left( {Rainy\left( d \right) \wedge \sim Cold\left( d \right)} \right)$$

$$\forall d\left( { \sim Rainy\left( d \right) \to Cold\left( d \right)} \right)$$

$$\exists d\left( { \sim Rainy\left( d \right) \to Cold\left( d \right)} \right)$$

$$\exists d\left( {Rainy\left( d \right) \wedge \sim Cold\left( d \right)} \right)$$

$$\left( {\left( {a \to b} \right) \wedge \left( {b \to c} \right)} \right) \to \left( {a \to c} \right)$$

$$\left( {a \leftrightarrow c} \right) \to \left( { \sim b \to \left( {a \wedge c} \right)} \right)$$

$$\left( {a \wedge b \wedge c} \right) \to \left( {c \vee a} \right)$$

$$A \to \left( {b \to a} \right)$$

$$\left( {\left( {p \leftrightarrow q} \right) \wedge r} \right) \vee \left( {p \wedge q \wedge \sim r} \right)$$

$$\left( { \sim \left( {p \leftrightarrow q} \right) \wedge r} \right) \vee \left( {p \wedge q \wedge \sim r} \right)$$

$$\left( {\left( {p \to q} \right) \wedge r} \right) \vee \left( {p \wedge q \wedge \sim r} \right)$$

$$\left( { \sim \left( {p \leftrightarrow q} \right) \wedge r} \right) \wedge \left( {p \wedge q \wedge \sim r} \right)$$

$$\exists x\left( {F\left( x \right) \wedge \neg P\left( x \right)} \right)$$

$$\exists x\left( {\neg F\left( x \right) \wedge P\left( x \right)} \right)$$

$$\exists x\left( {\neg F\left( x \right) \wedge \neg P\left( x \right)} \right)$$

$$\neg \exists x\left( {F\left( x \right) \wedge P\left( x \right)} \right)$$