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Theorem exlimd 1900
Description: Deduction form of 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 1868 . 2  |-  ( ph  ->  ( E. x ps 
->  E. x ch )
)
4 exlimd.2 . . 3  |-  F/ x ch
5419.9 1879 . 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 1599   F/wnf 1603
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-12 1840
This theorem depends on definitions:  df-bi 185  df-ex 1600  df-nf 1604
This theorem is referenced by:  exlimdh  1901  exlimdd  1966  equs5  2078  moexex  2349  2eu6  2369  exists2  2375  ceqsalgALT  3121  alxfr  4650  copsex2t  4724  mosubopt  4735  ovmpt2df  6419  ov3  6424  tz7.48-1  7110  ac6c4  8864  fsum2dlem  13564  gsum2d2lem  16875  fprod2dlem  29085  wl-lem-moexsb  29992  exlimddvf  30501  stoweidlem27  31698  fourierdlem31  31809  bj-equs5v  34080
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