Theorem reu6dv

Description: A condition which implies existential uniqueness. (Contributed by Thierry Arnoux, 13-Oct-2025)

	Ref	Expression
Hypotheses	reu6d.1	$$⊢\phi \to B\in A$$
	reu6d.2	$$⊢\left(\phi \wedge x\in A\right)\to \left(\psi \leftrightarrow x=B\right)$$
Assertion	reu6dv	$$⊢\phi \to \exists !x\in A\psi$$

Proof

Step	Hyp	Ref	Expression
1		reu6d.1	$$⊢\phi \to B\in A$$
2		reu6d.2	$$⊢\left(\phi \wedge x\in A\right)\to \left(\psi \leftrightarrow x=B\right)$$
3	2	ralrimiva	$$⊢\phi \to \forall x\in A\left(\psi \leftrightarrow x=B\right)$$
4		reu6i	$$⊢\left(B\in A\wedge \forall x\in A\left(\psi \leftrightarrow x=B\right)\right)\to \exists !x\in A\psi$$
5	1 3 4	syl2anc	$$⊢\phi \to \exists !x\in A\psi$$