Theorem alsi2d 13958

Description: Deduction rule: Given "all some" applied to a top-level inference, you can extract the "exists" part. (Contributed by David A. Wheeler, 20-Oct-2018.)

Hypothesis

Ref	Expression
alsi2d.1	|- ( ph -> A.! x ( ps -> ch ) )

Assertion

Ref	Expression
alsi2d	|- ( ph -> E. x ps )

Proof of Theorem alsi2d

Step	Hyp	Ref	Expression
1		alsi2d.1	. . 3 |- ( ph -> A.! x ( ps -> ch ) )
2		df-alsi 13954	. . 3 |- ( A.! x ( ps -> ch ) <-> ( A. x ( ps -> ch ) /\ E. x ps ) )
3	1, 2	sylib 121	. 2 |- ( ph -> ( A. x ( ps -> ch ) /\ E. x ps ) )
4	3	simprd 113	1 |- ( ph -> E. x ps )

Syntax hints:   -> wi 4   /\ wa 103   A.wal 1341   E.wex 1480   A.!walsi 13952

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106

This theorem depends on definitions:  df-bi 116  df-alsi 13954

This theorem is referenced by: (None)