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Theorem exlimd 2218
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 2216 . 2 (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))
4 exlimd.2 . . 3 𝑥𝜒
5419.9 2205 . 2 (∃𝑥𝜒𝜒)
63, 5imbitrdi 251 1 (𝜑 → (∃𝑥𝜓𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wex 1779  wnf 1783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-12 2177
This theorem depends on definitions:  df-bi 207  df-ex 1780  df-nf 1784
This theorem is referenced by:  exlimimdd  2219  exlimdh  2290  equs5  2465  moexexlem  2626  2eu6  2657  ceqsalgALT  3518  alxfr  5407  copsex2t  5497  mosubopt  5515  ov3  7596  tz7.48-1  8483  ac6c4  10521  fsum2dlem  15806  fprod2dlem  16016  gsum2d2lem  19991  exlimim  37343  exellim  37345  wl-lem-moexsb  37569  exlimddvf  38128  mnringmulrcld  44247  fourierdlem31  46153  or2expropbi  47046  ich2exprop  47458  ichreuopeq  47460  reuopreuprim  47513
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