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Theorem exlimd 2071
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 2069 . 2 (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))
4 exlimd.2 . . 3 𝑥𝜒
5419.9 2057 . 2 (∃𝑥𝜒𝜒)
63, 5syl6ib 239 1 (𝜑 → (∃𝑥𝜓𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wex 1694  wnf 1698
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-12 2031
This theorem depends on definitions:  df-bi 195  df-ex 1695  df-nf 1700
This theorem is referenced by:  exlimdd  2072  exlimdh  2131  equs5  2334  moexex  2524  2eu6  2541  exists2  2545  ceqsalgALT  3199  alxfr  4795  copsex2t  4873  mosubopt  4884  ovmpt2df  6664  ov3  6669  tz7.48-1  7398  ac6c4  9159  fsum2dlem  14285  fprod2dlem  14491  gsum2d2lem  18137  padct  28687  bj-equs5v  31745  exlimim  32164  exellim  32167  wl-lem-moexsb  32328  exlimddvf  32895  stoweidlem27  38720  fourierdlem31  38831  intsaluni  39023  isomenndlem  39220
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