MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  exlimd Structured version   Visualization version   GIF version

Theorem exlimd 2125
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 2123 . 2 (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))
4 exlimd.2 . . 3 𝑥𝜒
5419.9 2110 . 2 (∃𝑥𝜒𝜒)
63, 5syl6ib 241 1 (𝜑 → (∃𝑥𝜓𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wex 1744  wnf 1748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-12 2087
This theorem depends on definitions:  df-bi 197  df-ex 1745  df-nf 1750
This theorem is referenced by:  exlimdd  2126  exlimdh  2187  equs5  2379  moexex  2570  2eu6  2587  exists2  2591  ceqsalgALT  3262  alxfr  4908  copsex2t  4986  mosubopt  5001  ovmpt2df  6834  ov3  6839  tz7.48-1  7583  ac6c4  9341  fsum2dlem  14545  fprod2dlem  14754  gsum2d2lem  18418  padct  29625  exlimim  33319  exellim  33322  wl-lem-moexsb  33480  exlimddvf  34056  stoweidlem27  40562  fourierdlem31  40673  intsaluni  40865  isomenndlem  41065
  Copyright terms: Public domain W3C validator