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Theorem exellim 32821
 Description: Closed form of exellimddv 32822. See also exlimim 32818 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 2025 . . 3 𝑥𝑥(𝑥𝐴𝜑)
2 nfv 1840 . . 3 𝑥𝜑
3 sp 2051 . . 3 (∀𝑥(𝑥𝐴𝜑) → (𝑥𝐴𝜑))
41, 2, 3exlimd 2085 . 2 (∀𝑥(𝑥𝐴𝜑) → (∃𝑥 𝑥𝐴𝜑))
54impcom 446 1 ((∃𝑥 𝑥𝐴 ∧ ∀𝑥(𝑥𝐴𝜑)) → 𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384  ∀wal 1478  ∃wex 1701   ∈ wcel 1987 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-10 2016  ax-12 2044 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1702  df-nf 1707 This theorem is referenced by:  exellimddv  32822
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