Theorem riotacl2

Description: Membership law for "the unique element in A such that ph ". (Contributed by NM, 21-Aug-2011) (Revised by Mario Carneiro, 23-Dec-2016)

	Ref	Expression
Assertion	riotacl2	$$⊢\exists !x\in A\phi \to \left(\iota x\in A|\phi \right)\in \left\{x\in A|\phi \right\}$$

Proof

Step	Hyp	Ref	Expression
1		df-reu	$$⊢\exists !x\in A\phi \leftrightarrow \exists !x\left(x\in A\wedge \phi \right)$$
2		iotacl	$$⊢\exists !x\left(x\in A\wedge \phi \right)\to \left(\iota x|\left(x\in A\wedge \phi \right)\right)\in \left\{x|\left(x\in A\wedge \phi \right)\right\}$$
3	1 2	sylbi	$$⊢\exists !x\in A\phi \to \left(\iota x|\left(x\in A\wedge \phi \right)\right)\in \left\{x|\left(x\in A\wedge \phi \right)\right\}$$
4		df-riota	$$⊢\left(\iota x\in A|\phi \right)=\left(\iota x|\left(x\in A\wedge \phi \right)\right)$$
5		df-rab	$$⊢\left\{x\in A|\phi \right\}=\left\{x|\left(x\in A\wedge \phi \right)\right\}$$
6	3 4 5	3eltr4g	$$⊢\exists !x\in A\phi \to \left(\iota x\in A|\phi \right)\in \left\{x\in A|\phi \right\}$$