Metamath Proof Explorer


Theorem alsi2d

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)

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

Proof

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