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Theorem exlimd 1861
Description: Deduction from Theorem 19.9 of [Margaris] p. 89. (Contributed by NM, 23-Jan-1993.) (Revised 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 1830 . 2  |-  ( ph  ->  ( E. x ps 
->  E. x ch )
)
4 exlimd.2 . . 3  |-  F/ x ch
5419.9 1841 . 2  |-  ( E. x ch  <->  ch )
63, 5syl6ib 226 1  |-  ( ph  ->  ( E. x ps 
->  ch ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   E.wex 1596   F/wnf 1599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-12 1803
This theorem depends on definitions:  df-bi 185  df-ex 1597  df-nf 1600
This theorem is referenced by:  exlimdh  1862  exlimdd  1929  equs5  2065  moexex  2371  2eu6  2393  exists2  2399  ceqsalgALT  3139  alxfr  4660  copsex2t  4734  mosubopt  4745  ovmpt2df  6416  ov3  6421  tz7.48-1  7105  ac6c4  8857  fsum2dlem  13544  gsum2d2lem  16792  fprod2dlem  28687  wl-lem-moexsb  29594  suprnmpt  31029  stoweidlem27  31327  fourierdlem31  31438  bj-equs5v  33413
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