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Theorem exlimd 2216
 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 2214 . 2 (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))
4 exlimd.2 . . 3 𝑥𝜒
5419.9 2203 . 2 (∃𝑥𝜒𝜒)
63, 5syl6ib 254 1 (𝜑 → (∃𝑥𝜓𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∃wex 1781  Ⅎwnf 1785 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-12 2175 This theorem depends on definitions:  df-bi 210  df-ex 1782  df-nf 1786 This theorem is referenced by:  exlimimdd  2217  exlimddOLD  2219  exlimdh  2294  equs5  2472  moexexlem  2647  2eu6  2678  ceqsalgALT  3446  alxfr  5276  copsex2t  5351  mosubopt  5369  ov3  7307  tz7.48-1  8089  ac6c4  9941  fsum2dlem  15173  fprod2dlem  15382  gsum2d2lem  19161  exlimim  35039  exellim  35041  wl-lem-moexsb  35249  exlimddvf  35839  mnringmulrcld  41309  fourierdlem31  43146  or2expropbi  43992  ich2exprop  44356  ichreuopeq  44358  reuopreuprim  44411
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