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

Theorem alexnim 1582
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 1580 . . 3 (∃𝑦 ¬ 𝜑 → ¬ ∀𝑦𝜑)
21alimi 1387 . 2 (∀𝑥𝑦 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑦𝜑)
3 alnex 1431 . 2 (∀𝑥 ¬ ∀𝑦𝜑 ↔ ¬ ∃𝑥𝑦𝜑)
42, 3sylib 120 1 (∀𝑥𝑦 ¬ 𝜑 → ¬ ∃𝑥𝑦𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wal 1285  wex 1424
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 1379  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-4 1443  ax-17 1462  ax-ial 1470
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-fal 1293  df-nf 1393
This theorem is referenced by:  nalset  3937  bj-nalset  11143
  Copyright terms: Public domain W3C validator