Equivalences with restrictions

Provided that $x$ does not occur in $\varphi$ (or that $x$ is bound by another quantifier in $\varphi$), then the following equivalences hold:

Simplification: $\exists x\varphi \equiv \forall x\varphi \equiv \varphi$

Regarding disjunction:

• $\exists x(\varphi \vee \phi (x))\equiv \varphi \vee \exists x\phi (x)$
• $\forall x(\varphi \vee \phi (x))\equiv \varphi \vee \forall x\phi (x)$

Regarding conjunction:

• $\exists x(\varphi \wedge \phi (x))\equiv \varphi \wedge \exists x\phi (x)$
• $\forall x(\varphi \wedge \phi (x))\equiv \varphi \wedge \forall x\phi (x)$

We can also show similar equivalences for implications:

• $\exists x(\varphi \to \phi (x))\equiv \varphi \to \exists x\phi (x)$
• $\forall x(\varphi \to \phi (x))\equiv \varphi \to \forall x\phi (x)$
• $\exists x(\phi (x)\to \varphi )\equiv \forall x\phi (x)\to \varphi$
• $\forall x(\phi (x)\to \varphi )\equiv \exists x\phi (x)\to \varphi$