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Theorem exlimim 32818
Description: Closed form of exlimimd 32819. (Contributed by ML, 17-Jul-2020.)
Assertion
Ref Expression
exlimim ((∃𝑥𝜑 ∧ ∀𝑥(𝜑𝜓)) → 𝜓)
Distinct variable group:   𝜓,𝑥
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem exlimim
StepHypRef Expression
1 nfa1 2025 . . 3 𝑥𝑥(𝜑𝜓)
2 nfv 1840 . . 3 𝑥𝜓
3 sp 2051 . . 3 (∀𝑥(𝜑𝜓) → (𝜑𝜓))
41, 2, 3exlimd 2085 . 2 (∀𝑥(𝜑𝜓) → (∃𝑥𝜑𝜓))
54impcom 446 1 ((∃𝑥𝜑 ∧ ∀𝑥(𝜑𝜓)) → 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wal 1478  wex 1701
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: (None)
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