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	Could not format assertion : No typesetting found for |- ( AE x e. A ( ph -> ps ) -> E. x e. A ps ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		df-rals	Could not format ( AE x e. A ( ph -> ps ) <-> ( A. x e. A ( ph -> ps ) /\ E. x e. A ph ) ) : No typesetting found for |- ( AE x e. A ( ph -> ps ) <-> ( A. x e. A ( ph -> ps ) /\ E. x e. A ph ) ) with typecode |-
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	Could not format ( AE x e. A ( ph -> ps ) -> E. x e. A ps ) : No typesetting found for |- ( AE x e. A ( ph -> ps ) -> E. x e. A ps ) with typecode |-