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Theorem exellim 33690
Description: Closed form of exellimddv 33691. See also exlimim 33688 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 2195 . . 3 𝑥𝑥(𝑥𝐴𝜑)
2 nfv 2010 . . 3 𝑥𝜑
3 sp 2217 . . 3 (∀𝑥(𝑥𝐴𝜑) → (𝑥𝐴𝜑))
41, 2, 3exlimd 2253 . 2 (∀𝑥(𝑥𝐴𝜑) → (∃𝑥 𝑥𝐴𝜑))
54impcom 397 1 ((∃𝑥 𝑥𝐴 ∧ ∀𝑥(𝑥𝐴𝜑)) → 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  wal 1651  wex 1875  wcel 2157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-10 2185  ax-12 2213
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-ex 1876  df-nf 1880
This theorem is referenced by:  exellimddv  33691
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