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Theorem exellim 34507
Description: Closed form of exellimddv 34508. See also exlimim 34505 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 2146 . . 3 𝑥𝑥(𝑥𝐴𝜑)
2 nfv 1906 . . 3 𝑥𝜑
3 sp 2172 . . 3 (∀𝑥(𝑥𝐴𝜑) → (𝑥𝐴𝜑))
41, 2, 3exlimd 2208 . 2 (∀𝑥(𝑥𝐴𝜑) → (∃𝑥 𝑥𝐴𝜑))
54impcom 408 1 ((∃𝑥 𝑥𝐴 ∧ ∀𝑥(𝑥𝐴𝜑)) → 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wal 1526  wex 1771  wcel 2105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-10 2136  ax-12 2167
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-ex 1772  df-nf 1776
This theorem is referenced by:  exellimddv  34508
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