Theorem ralsex

Description: The consequent of an "all some" restricted to a class is witnessed: some member of A satisfying ph also satisfies ps . Restricted counterpart of alsex . (Contributed by David A. Wheeler, 12-Jul-2026)

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

Proof

Step	Hyp	Ref	Expression
1		df-rals	$$⊢\forall \exists x\in A\left(\phi \to \psi \right)\leftrightarrow \left(\forall x\in A\left(\phi \to \psi \right)\wedge \exists x\in A\phi \right)$$
2		rexim	$$⊢\forall x\in A\left(\phi \to \psi \right)\to \left(\exists x\in A\phi \to \exists x\in A\psi \right)$$
3	2	imp	$$⊢\left(\forall x\in A\left(\phi \to \psi \right)\wedge \exists x\in A\phi \right)\to \exists x\in A\psi$$
4	1 3	sylbi	$$⊢\forall \exists x\in A\left(\phi \to \psi \right)\to \exists x\in A\psi$$