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Theorem exlimd 2211
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 2209 . 2 (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))
4 exlimd.2 . . 3 𝑥𝜒
5419.9 2198 . 2 (∃𝑥𝜒𝜒)
63, 5syl6ib 252 1 (𝜑 → (∃𝑥𝜓𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wex 1773  wnf 1777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-12 2169
This theorem depends on definitions:  df-bi 208  df-ex 1774  df-nf 1778
This theorem is referenced by:  exlimimdd  2212  exlimddOLD  2214  exlimdh  2292  equs5  2480  moexexlem  2710  2eu6  2743  exists2OLD  2750  ceqsalgALT  3536  alxfr  5304  copsex2t  5380  mosubopt  5397  ov3  7301  tz7.48-1  8070  ac6c4  9892  fsum2dlem  15115  fprod2dlem  15324  gsum2d2lem  19013  exlimim  34492  exellim  34494  wl-lem-moexsb  34671  exlimddvf  35267  fourierdlem31  42289  or2expropbi  43135  ich2exprop  43465  ichreuopeq  43467  reuopreuprim  43520
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