Theorem rabsneu

Description: Restricted existential uniqueness determined by a singleton. (Contributed by NM, 29-May-2006) (Revised by Mario Carneiro, 23-Dec-2016)

	Ref	Expression
Assertion	rabsneu	$$⊢\left(A\in V\wedge \left\{x\in B|\phi \right\}=\left\{A\right\}\right)\to \exists !x\in B\phi$$

Proof

Step	Hyp	Ref	Expression
1		df-rab	$$⊢\left\{x\in B|\phi \right\}=\left\{x|\left(x\in B\wedge \phi \right)\right\}$$
2	1	eqeq1i	$$⊢\left\{x\in B|\phi \right\}=\left\{A\right\}\leftrightarrow \left\{x|\left(x\in B\wedge \phi \right)\right\}=\left\{A\right\}$$
3		absneu	$$⊢\left(A\in V\wedge \left\{x|\left(x\in B\wedge \phi \right)\right\}=\left\{A\right\}\right)\to \exists !x\left(x\in B\wedge \phi \right)$$
4	2 3	sylan2b	$$⊢\left(A\in V\wedge \left\{x\in B|\phi \right\}=\left\{A\right\}\right)\to \exists !x\left(x\in B\wedge \phi \right)$$
5		df-reu	$$⊢\exists !x\in B\phi \leftrightarrow \exists !x\left(x\in B\wedge \phi \right)$$
6	4 5	sylibr	$$⊢\left(A\in V\wedge \left\{x\in B|\phi \right\}=\left\{A\right\}\right)\to \exists !x\in B\phi$$