Theorem alsex 50456

Description: The consequent of an "all some" is witnessed: if 𝜓 holds of every 𝑥 satisfying 𝜑, and some 𝑥 satisfies 𝜑, then some 𝑥 satisfies 𝜓. This is the positive counterpart of als-no-surprise 50464, and it is the property that ordinary "for all" with implication lacks: from ∀𝑥(𝜑 → 𝜓) alone nothing whatever follows about 𝜓, as alimp-surprise 50438 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.)

Assertion

Ref	Expression
alsex	⊢ (∀∃𝑥(𝜑 → 𝜓) → ∃𝑥𝜓)

Proof of Theorem alsex

Step	Hyp	Ref	Expression
1		df-als 50446	. 2 ⊢ (∀∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥(𝜑 → 𝜓) ∧ ∃𝑥𝜑))
2		exim 1861	. . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓))
3	2	imp 411	. 2 ⊢ ((∀𝑥(𝜑 → 𝜓) ∧ ∃𝑥𝜑) → ∃𝑥𝜓)
4	1, 3	sylbi 220	1 ⊢ (∀∃𝑥(𝜑 → 𝜓) → ∃𝑥𝜓)

Syntax hints:   → wi 4   ∧ wa 400  ∀wal 1565  ∃wex 1806  ∀∃wals 50444

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807  df-als 50446

This theorem is referenced by: (None)