# Question about equivalencies when using the existential quantifier

I'm currently in a boolean algebra class, and we are asked if the statement:

$$\exists xM(x) \wedge \exists xD(x)$$

is a proposition. Although I know that it is a proposition, I was wondering if anyone knew if it could be simplified to

$$\exists x (M(x) \wedge D(x))$$

Since I know that this is true at least:

$$\forall x(M(x) \wedge D(x)) \equiv \forall x M(x) \wedge \forall x D(x)$$

• A rule you can use is : $$\exists x~\bigg(M(x) \color{red}{\lor} D(x)\bigg)\quad \equiv \quad \bigg(\exists x ~M(x)\bigg) \color{red}{\lor} \bigg(\exists x ~D(x)\bigg)$$ Jan 16, 2015 at 23:26

Perhaps let $M$ be "Man who likes cookies." Let $D$ be "Woman who likes dogs."
The first statement $$\exists xM(x) \wedge \exists xD(x)$$ means that there is a man who likes cookies (Me) and there is a woman who likes dogs (My wife).
But, the second statement $$\exists x(M(x) \wedge D(x))$$ means "There exists a person that is both a man who likes cookies and a woman that likes dogs." Less likely. The two statements are not equivalent.