Theorem ralsex 17048

Description: The consequent of an "all some" restricted to a class is witnessed: some member of A satisfying ph also satisfies ps. Restricted counterpart of alsex 17047. (Contributed by David A. Wheeler, 12-Jul-2026.)

Assertion

Ref	Expression
ralsex	|- ( A.E. x e. A ( ph -> ps ) -> E. x e. A ps )

Proof of Theorem ralsex

Step	Hyp	Ref	Expression
1		df-rals 17037	. 2 |- ( A.E. x e. A ( ph -> ps ) <-> ( A. x e. A ( ph -> ps ) /\ E. x e. A ph ) )
2		rexim 2644	. . 3 |- ( A. x e. A ( ph -> ps ) -> ( E. x e. A ph -> E. x e. A ps ) )
3	2	imp 124	. 2 |- ( ( A. x e. A ( ph -> ps ) /\ E. x e. A ph ) -> E. x e. A ps )
4	1, 3	sylbi 121	1 |- ( A.E. x e. A ( ph -> ps ) -> E. x e. A ps )

Syntax hints:   -> wi 4   /\ wa 104   A.wral 2528   E.wrex 2529   A.E.wrals 17035

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-ral 2533  df-rex 2534  df-rals 17037

This theorem is referenced by: (None)