Theorem riotaclbBAD

Description: Closure of restricted iota. (Contributed by NM, 15-Sep-2011)

	Ref	Expression
Hypothesis	riotaclb.1	$$⊢A\in \mathrm{V}$$
Assertion	riotaclbBAD	$$⊢\exists !x\in A\phi \leftrightarrow \left(\iota x\in A|\phi \right)\in A$$

Proof

Step	Hyp	Ref	Expression
1		riotaclb.1	$$⊢A\in \mathrm{V}$$
2		riotaclbgBAD	$$⊢A\in \mathrm{V}\to \left(\exists !x\in A\phi \leftrightarrow \left(\iota x\in A|\phi \right)\in A\right)$$
3	1 2	ax-mp	$$⊢\exists !x\in A\phi \leftrightarrow \left(\iota x\in A|\phi \right)\in A$$