Users' Mathboxes Mathbox for David A. Wheeler < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  alsi2d Unicode version

Theorem alsi2d 11060
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
StepHypRef Expression
1 alsi2d.1 . . 3  |-  ( ph  ->  A.! x ( ps 
->  ch ) )
2 df-alsi 11056 . . 3  |-  ( A.! x ( ps  ->  ch )  <->  ( A. x
( ps  ->  ch )  /\  E. x ps ) )
31, 2sylib 120 . 2  |-  ( ph  ->  ( A. x ( ps  ->  ch )  /\  E. x ps )
)
43simprd 112 1  |-  ( ph  ->  E. x ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102   A.wal 1283   E.wex 1422   A.!walsi 11054
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105
This theorem depends on definitions:  df-bi 115  df-alsi 11056
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator