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Theorem ralexim 2361
Description: Relationship between restricted universal and existential quantifiers. (Contributed by Jim Kingdon, 17-Aug-2018.)
Assertion
Ref Expression
ralexim (∀𝑥𝐴 𝜑 → ¬ ∃𝑥𝐴 ¬ 𝜑)

Proof of Theorem ralexim
StepHypRef Expression
1 rexnalim 2360 . 2 (∃𝑥𝐴 ¬ 𝜑 → ¬ ∀𝑥𝐴 𝜑)
21con2i 590 1 (∀𝑥𝐴 𝜑 → ¬ ∃𝑥𝐴 ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wral 2349  wrex 2350
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-ie1 1423  ax-ie2 1424  ax-4 1441  ax-17 1460  ax-ial 1468
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291  df-nf 1391  df-ral 2354  df-rex 2355
This theorem is referenced by: (None)
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