Theorem aiotaexaiotaiota

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

	Ref	Expression
Assertion	aiotaexaiotaiota	$$⊢\iota \in \mathrm{V}\leftrightarrow \left(\iota x|\phi \right)=\iota$$

Proof

Step	Hyp	Ref	Expression
1		aiotaexb	$$⊢\exists !x\phi \leftrightarrow \iota \in \mathrm{V}$$
2		reuaiotaiota	$$⊢\exists !x\phi \leftrightarrow \left(\iota x|\phi \right)=\iota$$
3	1 2	bitr3i	$$⊢\iota \in \mathrm{V}\leftrightarrow \left(\iota x|\phi \right)=\iota$$