Theorem alsex

Description: The consequent of an "all some" is witnessed: if ps holds of every x satisfying ph , and some x satisfies ph , then some x satisfies ps . This is the positive counterpart of als-no-surprise , and it is the property that ordinary "for all" with implication lacks: from A. x ( ph -> ps ) alone nothing whatever follows about ps , as alimp-surprise shows. It is the reason the allsome quantifier says what a speaker of "all Martians are green" usually means. (Contributed by David A. Wheeler, 12-Jul-2026)

	Ref	Expression
Assertion	alsex	|- ( AE x ( ph -> ps ) -> E. x ps )

Proof

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