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Theorem exlimd 1824
Description: Deduction from Theorem 19.9 of [Margaris] p. 89. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 12-Jan-2018.)
Hypotheses
Ref Expression
exlimd.1  |-  F/ x ph
exlimd.2  |-  F/ x ch
exlimd.3  |-  ( ph  ->  ( ps  ->  ch ) )
Assertion
Ref Expression
exlimd  |-  ( ph  ->  ( E. x ps 
->  ch ) )

Proof of Theorem exlimd
StepHypRef Expression
1 exlimd.1 . . 3  |-  F/ x ph
2 exlimd.3 . . 3  |-  ( ph  ->  ( ps  ->  ch ) )
31, 2eximd 1786 . 2  |-  ( ph  ->  ( E. x ps 
->  E. x ch )
)
4 exlimd.2 . . 3  |-  F/ x ch
5419.9 1797 . 2  |-  ( E. x ch  <->  ch )
63, 5syl6ib 218 1  |-  ( ph  ->  ( E. x ps 
->  ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   E.wex 1550   F/wnf 1553
This theorem is referenced by:  exlimdh  1826  exlimdd  1912  equs5  2085  exists2  2370  ceqsalg  2967  copsex2t  4430  mosubopt  4441  alxfr  4722  ovmpt2df  6191  ov3  6196  tz7.48-1  6686  ac6c4  8345  fsum2dlem  12537  gsum2d2lem  15530  fprod2dlem  25288  stoweidlem27  27685
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-11 1761
This theorem depends on definitions:  df-bi 178  df-ex 1551  df-nf 1554
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