Theorem mobi

Description: Equivalence theorem for the at-most-one quantifier. (Contributed by BJ, 7-Oct-2022) (Proof shortened by Wolf Lammen, 18-Feb-2023)

	Ref	Expression
Assertion	mobi	$$⊢\forall x\left(\phi \leftrightarrow \psi \right)\to \left({\exists }^{*}x\phi \leftrightarrow {\exists }^{*}x\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		albiim	$$⊢\forall x\left(\phi \leftrightarrow \psi \right)\leftrightarrow \left(\forall x\left(\phi \to \psi \right)\wedge \forall x\left(\psi \to \phi \right)\right)$$
2		moim	$$⊢\forall x\left(\psi \to \phi \right)\to \left({\exists }^{*}x\phi \to {\exists }^{*}x\psi \right)$$
3		moim	$$⊢\forall x\left(\phi \to \psi \right)\to \left({\exists }^{*}x\psi \to {\exists }^{*}x\phi \right)$$
4	2 3	impbid21d	$$⊢\forall x\left(\phi \to \psi \right)\to \left(\forall x\left(\psi \to \phi \right)\to \left({\exists }^{*}x\phi \leftrightarrow {\exists }^{*}x\psi \right)\right)$$
5	4	imp	$$⊢\left(\forall x\left(\phi \to \psi \right)\wedge \forall x\left(\psi \to \phi \right)\right)\to \left({\exists }^{*}x\phi \leftrightarrow {\exists }^{*}x\psi \right)$$
6	1 5	sylbi	$$⊢\forall x\left(\phi \leftrightarrow \psi \right)\to \left({\exists }^{*}x\phi \leftrightarrow {\exists }^{*}x\psi \right)$$