Theorem ralseu2d

Description: Deduction rule: Given "all some one" applied to a class, you can extract the "exactly one" part. Note that the witness must satisfy the antecedent ps , not merely be a member of A . (Contributed by David A. Wheeler, 21-Jul-2026)

	Ref	Expression
Hypothesis	ralseu2d.1	$$⊢\phi \to \forall \exists !x\in A\left(\psi \to \chi \right)$$
Assertion	ralseu2d	$$⊢\phi \to \exists !x\in A\psi$$

Proof

Step	Hyp	Ref	Expression
1		ralseu2d.1	$$⊢\phi \to \forall \exists !x\in A\left(\psi \to \chi \right)$$
2		df-ralseu	$$⊢\forall \exists !x\in A\left(\psi \to \chi \right)\leftrightarrow \left(\forall x\in A\left(\psi \to \chi \right)\wedge \exists !x\in A\psi \right)$$
3	1 2	sylib	$$⊢\phi \to \left(\forall x\in A\left(\psi \to \chi \right)\wedge \exists !x\in A\psi \right)$$
4	3	simprd	$$⊢\phi \to \exists !x\in A\psi$$