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	$$⊢\forall \exists x\left(\phi \to \psi \right)\to \exists x\psi$$

Proof

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