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Theorem exlimd 2251
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 2249 . 2 (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))
4 exlimd.2 . . 3 𝑥𝜒
5419.9 2237 . 2 (∃𝑥𝜒𝜒)
63, 5syl6ib 242 1 (𝜑 → (∃𝑥𝜓𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wex 1874  wnf 1878
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-12 2211
This theorem depends on definitions:  df-bi 198  df-ex 1875  df-nf 1879
This theorem is referenced by:  exlimdd  2252  exlimdh  2324  equs5  2441  moexex  2663  2eu6  2680  exists2  2685  ceqsalgALT  3384  alxfr  5042  copsex2t  5112  mosubopt  5131  ov3  6995  tz7.48-1  7742  ac6c4  9556  fsum2dlem  14786  fprod2dlem  14993  gsum2d2lem  18638  exlimim  33623  exellim  33625  wl-lem-moexsb  33775  exlimddvf  34347  fourierdlem31  40992
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