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	|- ( AE x e. A ( ph -> ps ) -> E. x e. A ps )

Proof

Step	Hyp	Ref	Expression
1		df-rals	|- ( AE x e. A ( ph -> ps ) <-> ( A. x e. A ( ph -> ps ) /\ E. x e. A ph ) )
2		rexim	|- ( A. x e. A ( ph -> ps ) -> ( E. x e. A ph -> E. x e. A ps ) )
3	2	imp	|- ( ( A. x e. A ( ph -> ps ) /\ E. x e. A ph ) -> E. x e. A ps )
4	1 3	sylbi	|- ( AE x e. A ( ph -> ps ) -> E. x e. A ps )