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	⊢ ( ∀∃ 𝑥 ( 𝜑 → 𝜓 ) → ∃ 𝑥 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		df-als	⊢ ( ∀∃ 𝑥 ( 𝜑 → 𝜓 ) ↔ ( ∀ 𝑥 ( 𝜑 → 𝜓 ) ∧ ∃ 𝑥 𝜑 ) )
2		exim	⊢ ( ∀ 𝑥 ( 𝜑 → 𝜓 ) → ( ∃ 𝑥 𝜑 → ∃ 𝑥 𝜓 ) )
3	2	imp	⊢ ( ( ∀ 𝑥 ( 𝜑 → 𝜓 ) ∧ ∃ 𝑥 𝜑 ) → ∃ 𝑥 𝜓 )
4	1 3	sylbi	⊢ ( ∀∃ 𝑥 ( 𝜑 → 𝜓 ) → ∃ 𝑥 𝜓 )