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Theorem rexalim 2316
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 2313 . . 3 (∀𝑥𝐴 ¬ 𝜑 ↔ ¬ ∃𝑥𝐴 𝜑)
21biimpi 113 . 2 (∀𝑥𝐴 ¬ 𝜑 → ¬ ∃𝑥𝐴 𝜑)
32con2i 557 1 (∃𝑥𝐴 𝜑 → ¬ ∀𝑥𝐴 ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wral 2303  wrex 2304
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1336  ax-gen 1338  ax-ie2 1383
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-ral 2308  df-rex 2309
This theorem is referenced by: (None)
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