ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  alexnim GIF version

Theorem alexnim 1648
Description: A relationship between two quantifiers and negation. (Contributed by Jim Kingdon, 27-Aug-2018.)
Assertion
Ref Expression
alexnim (∀𝑥𝑦 ¬ 𝜑 → ¬ ∃𝑥𝑦𝜑)

Proof of Theorem alexnim
StepHypRef Expression
1 exnalim 1646 . . 3 (∃𝑦 ¬ 𝜑 → ¬ ∀𝑦𝜑)
21alimi 1455 . 2 (∀𝑥𝑦 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑦𝜑)
3 alnex 1499 . 2 (∀𝑥 ¬ ∀𝑦𝜑 ↔ ¬ ∃𝑥𝑦𝜑)
42, 3sylib 122 1 (∀𝑥𝑦 ¬ 𝜑 → ¬ ∃𝑥𝑦𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wal 1351  wex 1492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-nf 1461
This theorem is referenced by:  nalset  4132  bj-nalset  14507
  Copyright terms: Public domain W3C validator