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Theorem exlimd 2085
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 2083 . 2 (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))
4 exlimd.2 . . 3 𝑥𝜒
5419.9 2070 . 2 (∃𝑥𝜒𝜒)
63, 5syl6ib 241 1 (𝜑 → (∃𝑥𝜓𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wex 1702  wnf 1706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-12 2045
This theorem depends on definitions:  df-bi 197  df-ex 1703  df-nf 1708
This theorem is referenced by:  exlimdd  2086  exlimdh  2147  equs5  2349  moexex  2539  2eu6  2556  exists2  2560  ceqsalgALT  3226  alxfr  4869  copsex2t  4947  mosubopt  4962  ovmpt2df  6777  ov3  6782  tz7.48-1  7523  ac6c4  9288  fsum2dlem  14482  fprod2dlem  14691  gsum2d2lem  18353  padct  29471  exlimim  33160  exellim  33163  wl-lem-moexsb  33321  exlimddvf  33897  stoweidlem27  40007  fourierdlem31  40118  intsaluni  40310  isomenndlem  40507
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