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Theorem rexalim 2367
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 2363 . . 3 (∀𝑥𝐴 ¬ 𝜑 ↔ ¬ ∃𝑥𝐴 𝜑)
21biimpi 118 . 2 (∀𝑥𝐴 ¬ 𝜑 → ¬ ∃𝑥𝐴 𝜑)
32con2i 590 1 (∃𝑥𝐴 𝜑 → ¬ ∀𝑥𝐴 ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wral 2353  wrex 2354
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-5 1377  ax-gen 1379  ax-ie2 1424
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291  df-ral 2358  df-rex 2359
This theorem is referenced by:  infnlbti  6628
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