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Theorem alsc2d 13439
Description: Deduction rule: Given "all some" applied to a class, you can extract the "there exists" part. (Contributed by David A. Wheeler, 20-Oct-2018.)
Hypothesis
Ref Expression
alsc2d.1  |-  ( ph  ->  A.! x  e.  A ps )
Assertion
Ref Expression
alsc2d  |-  ( ph  ->  E. x  x  e.  A )

Proof of Theorem alsc2d
StepHypRef Expression
1 alsc2d.1 . . 3  |-  ( ph  ->  A.! x  e.  A ps )
2 df-alsc 13434 . . 3  |-  ( A.! x  e.  A ps  <->  ( A. x  e.  A  ps  /\  E. x  x  e.  A ) )
31, 2sylib 121 . 2  |-  ( ph  ->  ( A. x  e.  A  ps  /\  E. x  x  e.  A
) )
43simprd 113 1  |-  ( ph  ->  E. x  x  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   E.wex 1469    e. wcel 1481   A.wral 2417   A.!walsc 13432
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-alsc 13434
This theorem is referenced by: (None)
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