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Theorem exlimd 2074
 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 2072 . 2 (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))
4 exlimd.2 . . 3 𝑥𝜒
5419.9 2060 . 2 (∃𝑥𝜒𝜒)
63, 5syl6ib 240 1 (𝜑 → (∃𝑥𝜓𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∃wex 1695  Ⅎwnf 1699 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034 This theorem depends on definitions:  df-bi 196  df-ex 1696  df-nf 1701 This theorem is referenced by:  exlimdd  2075  exlimdh  2134  equs5  2339  moexex  2529  2eu6  2546  exists2  2550  ceqsalgALT  3204  alxfr  4804  copsex2t  4883  mosubopt  4897  ovmpt2df  6690  ov3  6695  tz7.48-1  7425  ac6c4  9186  fsum2dlem  14343  fprod2dlem  14549  gsum2d2lem  18195  padct  28885  exlimim  32365  exellim  32368  wl-lem-moexsb  32529  exlimddvf  33096  stoweidlem27  38920  fourierdlem31  39031  intsaluni  39223  isomenndlem  39420
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