Theorem alsex 17047

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 17055, and it is the property that ordinary "for all" with implication lacks: from A. x ( ph -> ps ) alone nothing whatever follows about ps, since nothing need satisfy ph. It is 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	|- ( A.E. x ( ph -> ps ) -> E. x ps )

Proof of Theorem alsex

Step	Hyp	Ref	Expression
1		df-als 17036	. 2 |- ( A.E. x ( ph -> ps ) <-> ( A. x ( ph -> ps ) /\ E. x ph ) )
2		exim 1652	. . 3 |- ( A. x ( ph -> ps ) -> ( E. x ph -> E. x ps ) )
3	2	imp 124	. 2 |- ( ( A. x ( ph -> ps ) /\ E. x ph ) -> E. x ps )
4	1, 3	sylbi 121	1 |- ( A.E. x ( ph -> ps ) -> E. x ps )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1400   E.wex 1545   A.E.wals 17034

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-ial 1587

This theorem depends on definitions:  df-bi 117  df-als 17036

This theorem is referenced by: (None)