Theorem ralseud

Description: Introduction rule for "all some one" restricted to a class. This is the converse of ralseu1d and ralseu2d taken together. (Contributed by David A. Wheeler, 21-Jul-2026)

	Ref	Expression
Hypotheses	ralseud.1	$$⊢\phi \to \forall x\in A\left(\psi \to \chi \right)$$
	ralseud.2	$$⊢\phi \to \exists !x\in A\psi$$
Assertion	ralseud	$$⊢\phi \to \forall \exists !x\in A\left(\psi \to \chi \right)$$

Proof

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