Theorem bj-moeub

Description: Uniqueness is equivalent to existence being equivalent to unique existence. (Contributed by BJ, 14-Oct-2022)

	Ref	Expression
Assertion	bj-moeub	$$⊢{\exists }^{*}x\phi \leftrightarrow \left(\exists x\phi \leftrightarrow \exists !x\phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		moeu	$$⊢{\exists }^{*}x\phi \leftrightarrow \left(\exists x\phi \to \exists !x\phi \right)$$
2		euex	$$⊢\exists !x\phi \to \exists x\phi$$
3		impbi	$$⊢\left(\exists x\phi \to \exists !x\phi \right)\to \left(\left(\exists !x\phi \to \exists x\phi \right)\to \left(\exists x\phi \leftrightarrow \exists !x\phi \right)\right)$$
4	2 3	mpi	$$⊢\left(\exists x\phi \to \exists !x\phi \right)\to \left(\exists x\phi \leftrightarrow \exists !x\phi \right)$$
5		biimp	$$⊢\left(\exists x\phi \leftrightarrow \exists !x\phi \right)\to \left(\exists x\phi \to \exists !x\phi \right)$$
6	4 5	impbii	$$⊢\left(\exists x\phi \to \exists !x\phi \right)\leftrightarrow \left(\exists x\phi \leftrightarrow \exists !x\phi \right)$$
7	1 6	bitri	$$⊢{\exists }^{*}x\phi \leftrightarrow \left(\exists x\phi \leftrightarrow \exists !x\phi \right)$$