information:classification
Conjunction and disjunction

The rules for conjunction and disjunction say that

and

The equivalences are valid when x {\displaystyle x} does not appear as a free variable of ψ {\displaystyle \psi }; if x {\displaystyle x} does appear free in ψ {\displaystyle \psi }, one can rename the bound x {\displaystyle x} in (∃ x ϕ) {\displaystyle (\exists x\phi)} and obtain the equivalent (∃ x ′ ϕ) {\displaystyle (\exists x'\phi)}.

For example, in the language of rings,

but

because the formula on the left is true in any ring when the free variable x is equal to 0, while the formula on the right has no free variables and is false in any nontrivial ring. So (∃ x (x 2 = 1)) ∧ (0 = x) {\displaystyle (\exists x(x^{2}=1))\land (0=x)} will be first rewritten as (∃ x ′ (x ′ 2 = 1)) ∧ (0 = x) {\displaystyle (\exists x'(x'^{2}=1))\land (0=x)} and then put in prenex normal form ∃ x ′ (x ′ 2 = 1 ∧ 0 = x) {\displaystyle \exists x'(x'^{2}=1\land 0=x)}.