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Theorem rexalim 2389
Description: Relationship between restricted universal and existential quantifiers. (Contributed by Jim Kingdon, 17-Aug-2018.)
Assertion
Ref Expression
rexalim (∃𝑥𝐴 𝜑 → ¬ ∀𝑥𝐴 ¬ 𝜑)

Proof of Theorem rexalim
StepHypRef Expression
1 ralnex 2385 . . 3 (∀𝑥𝐴 ¬ 𝜑 ↔ ¬ ∃𝑥𝐴 𝜑)
21biimpi 119 . 2 (∀𝑥𝐴 ¬ 𝜑 → ¬ ∃𝑥𝐴 𝜑)
32con2i 597 1 (∃𝑥𝐴 𝜑 → ¬ ∀𝑥𝐴 ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wral 2375  wrex 2376
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-5 1391  ax-gen 1393  ax-ie2 1438
This theorem depends on definitions:  df-bi 116  df-tru 1302  df-fal 1305  df-ral 2380  df-rex 2381
This theorem is referenced by:  infnlbti  6828
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