Theorem iotaint

Description: Equivalence between two different forms of iota . (Contributed by Mario Carneiro, 24-Dec-2016)

	Ref	Expression
Assertion	iotaint	$$⊢\exists !x\phi \to \left(\iota x|\phi \right)=\bigcap \left\{x|\phi \right\}$$

Proof

Step	Hyp	Ref	Expression
1		iotauni	$$⊢\exists !x\phi \to \left(\iota x|\phi \right)=\bigcup \left\{x|\phi \right\}$$
2		uniintab	$$⊢\exists !x\phi \leftrightarrow \bigcup \left\{x|\phi \right\}=\bigcap \left\{x|\phi \right\}$$
3	2	biimpi	$$⊢\exists !x\phi \to \bigcup \left\{x|\phi \right\}=\bigcap \left\{x|\phi \right\}$$
4	1 3	eqtrd	$$⊢\exists !x\phi \to \left(\iota x|\phi \right)=\bigcap \left\{x|\phi \right\}$$