Theorem alsd 17039

Description: Introduction rule: "all some" holds if the "for all" part holds and the antecedent has a witness. This is the converse of als1d 17041 and als2d 17042 taken together, and is what lets an "all some" statement be proved rather than merely taken apart. (Contributed by David A. Wheeler, 12-Jul-2026.)

Hypotheses

Ref	Expression
alsd.1	|- ( ph -> A. x ( ps -> ch ) )
alsd.2	|- ( ph -> E. x ps )

Assertion

Ref	Expression
alsd	|- ( ph -> A.E. x ( ps -> ch ) )

Proof of Theorem alsd

Step	Hyp	Ref	Expression
1		alsd.1	. 2 |- ( ph -> A. x ( ps -> ch ) )
2		alsd.2	. 2 |- ( ph -> E. x ps )
3		df-als 17036	. 2 |- ( A.E. x ( ps -> ch ) <-> ( A. x ( ps -> ch ) /\ E. x ps ) )
4	1, 2, 3	sylanbrc 421	1 |- ( ph -> A.E. x ( ps -> ch ) )

Syntax hints:   -> wi 4   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

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

This theorem is referenced by: (None)