Users' Mathboxes Mathbox for ML < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  exellim Structured version   Visualization version   GIF version

Theorem exellim 34619
Description: Closed form of exellimddv 34620. See also exlimim 34617 for a more general theorem. (Contributed by ML, 17-Jul-2020.)
Assertion
Ref Expression
exellim ((∃𝑥 𝑥𝐴 ∧ ∀𝑥(𝑥𝐴𝜑)) → 𝜑)
Distinct variable group:   𝜑,𝑥
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem exellim
StepHypRef Expression
1 nfa1 2151 . . 3 𝑥𝑥(𝑥𝐴𝜑)
2 nfv 1911 . . 3 𝑥𝜑
3 sp 2177 . . 3 (∀𝑥(𝑥𝐴𝜑) → (𝑥𝐴𝜑))
41, 2, 3exlimd 2213 . 2 (∀𝑥(𝑥𝐴𝜑) → (∃𝑥 𝑥𝐴𝜑))
54impcom 410 1 ((∃𝑥 𝑥𝐴 ∧ ∀𝑥(𝑥𝐴𝜑)) → 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  wal 1531  wex 1776  wcel 2110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-10 2141  ax-12 2172
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1777  df-nf 1781
This theorem is referenced by:  exellimddv  34620
  Copyright terms: Public domain W3C validator