Theorem reu6i

Description: A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015)

	Ref	Expression
Assertion	reu6i	$$⊢\left(B\in A\wedge \forall x\in A\left(\phi \leftrightarrow x=B\right)\right)\to \exists !x\in A\phi$$

Proof

Step	Hyp	Ref	Expression
1		eqeq2	$$⊢y=B\to \left(x=y\leftrightarrow x=B\right)$$
2	1	bibi2d	$$⊢y=B\to \left(\left(\phi \leftrightarrow x=y\right)\leftrightarrow \left(\phi \leftrightarrow x=B\right)\right)$$
3	2	ralbidv	$$⊢y=B\to \left(\forall x\in A\left(\phi \leftrightarrow x=y\right)\leftrightarrow \forall x\in A\left(\phi \leftrightarrow x=B\right)\right)$$
4	3	rspcev	$$⊢\left(B\in A\wedge \forall x\in A\left(\phi \leftrightarrow x=B\right)\right)\to \exists y\in A\forall x\in A\left(\phi \leftrightarrow x=y\right)$$
5		reu6	$$⊢\exists !x\in A\phi \leftrightarrow \exists y\in A\forall x\in A\left(\phi \leftrightarrow x=y\right)$$
6	4 5	sylibr	$$⊢\left(B\in A\wedge \forall x\in A\left(\phi \leftrightarrow x=B\right)\right)\to \exists !x\in A\phi$$