Theorem ralseurals

Description: "All some one" restricted to a class implies "all some" restricted to that class. Restricted counterpart of alseuals . (Contributed by David A. Wheeler, 21-Jul-2026)

	Ref	Expression
Assertion	ralseurals	$$⊢\forall \exists !x\in A\left(\phi \to \psi \right)\to \forall \exists x\in A\left(\phi \to \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		reurex	$$⊢\exists !x\in A\phi \to \exists x\in A\phi$$
2	1	anim2i	$$⊢\left(\forall x\in A\left(\phi \to \psi \right)\wedge \exists !x\in A\phi \right)\to \left(\forall x\in A\left(\phi \to \psi \right)\wedge \exists x\in A\phi \right)$$
3		df-ralseu	$$⊢\forall \exists !x\in A\left(\phi \to \psi \right)\leftrightarrow \left(\forall x\in A\left(\phi \to \psi \right)\wedge \exists !x\in A\phi \right)$$
4		df-rals	$$⊢\forall \exists x\in A\left(\phi \to \psi \right)\leftrightarrow \left(\forall x\in A\left(\phi \to \psi \right)\wedge \exists x\in A\phi \right)$$
5	2 3 4	3imtr4i	$$⊢\forall \exists !x\in A\left(\phi \to \psi \right)\to \forall \exists x\in A\left(\phi \to \psi \right)$$