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Theorem exlimim 34610
Description: Closed form of exlimimd 34611. (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 2148 . . 3 𝑥𝑥(𝜑𝜓)
2 nfv 1908 . . 3 𝑥𝜓
3 sp 2174 . . 3 (∀𝑥(𝜑𝜓) → (𝜑𝜓))
41, 2, 3exlimd 2210 . 2 (∀𝑥(𝜑𝜓) → (∃𝑥𝜑𝜓))
54impcom 410 1 ((∃𝑥𝜑 ∧ ∀𝑥(𝜑𝜓)) → 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  wal 1528  wex 1773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-10 2138  ax-12 2169
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1774  df-nf 1778
This theorem is referenced by: (None)
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