Theorem reuaiotaiota

Description: The iota and the alternate iota over a wff ph are equal iff there is a unique value x satisfying ph . (Contributed by AV, 25-Aug-2022)

	Ref	Expression
Assertion	reuaiotaiota	$$⊢\exists !x\phi \leftrightarrow \left(\iota x|\phi \right)=\iota$$

Proof

Step	Hyp	Ref	Expression
1		euabsneu	$$⊢\exists !x\phi \leftrightarrow \exists !y\left\{x|\phi \right\}=\left\{y\right\}$$
2		reuabaiotaiota	$$⊢\exists !y\left\{x|\phi \right\}=\left\{y\right\}\leftrightarrow \left(\iota x|\phi \right)=\iota$$
3	1 2	bitri	$$⊢\exists !x\phi \leftrightarrow \left(\iota x|\phi \right)=\iota$$