# Give an equational proof $\vdash (\exists x)(A \lor B) \equiv (\exists x)A \lor (\exists x)B$

Give an equational proof $$\vdash (\exists x)(A \lor B) \equiv (\exists x)A \lor (\exists x)B$$

What I tried

$(\exists x)(A \lor B)$

Applying Definition of $\exists$

$\lnot (\forall x)\lnot (A \lor B)$

Applying De morgan

$\lnot (\forall x) (\lnot A \land \lnot B)$

Applying Distributivity of $\forall$ over $\land$

$\lnot (\forall x) \lnot A \land \lnot(\forall x) \lnot B$

Applying Definition of $\exists$

$(\exists x)A \land (\exists x)B$

What can I do next ?

See George Tourlakis, Mathematical Logic (2008) or this post for a list of axioms and theorems.

$$¬(∀x)(¬A ∧ ¬B)$$
$$¬((∀x)¬A ∧ (∀x)¬B)$$
$$¬(¬(\exists x)A ∧ ¬(\exists x)B)$$
$$(\exists x)A \lor (\exists x)B$$