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Theorem exlimd 2208
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 𝑥𝜑
exlimd.2 𝑥𝜒
exlimd.3 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
exlimd (𝜑 → (∃𝑥𝜓𝜒))

Proof of Theorem exlimd
StepHypRef Expression
1 exlimd.1 . . 3 𝑥𝜑
2 exlimd.3 . . 3 (𝜑 → (𝜓𝜒))
31, 2eximd 2206 . 2 (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))
4 exlimd.2 . . 3 𝑥𝜒
5419.9 2195 . 2 (∃𝑥𝜒𝜒)
63, 5syl6ib 252 1 (𝜑 → (∃𝑥𝜓𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wex 1771  wnf 1775
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-12 2167
This theorem depends on definitions:  df-bi 208  df-ex 1772  df-nf 1776
This theorem is referenced by:  exlimimdd  2209  exlimddOLD  2211  exlimdh  2289  equs5  2475  moexexlem  2704  2eu6  2737  ceqsalgALT  3528  alxfr  5298  copsex2t  5374  mosubopt  5391  ov3  7300  tz7.48-1  8068  ac6c4  9891  fsum2dlem  15113  fprod2dlem  15322  gsum2d2lem  19022  exlimim  34505  exellim  34507  wl-lem-moexsb  34685  exlimddvf  35280  fourierdlem31  42300  or2expropbi  43146  ich2exprop  43510  ichreuopeq  43512  reuopreuprim  43565
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