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

Theorem exanaliim 1627
 Description: A transformation of quantifiers and logical connectives. In classical logic the converse also holds. (Contributed by Jim Kingdon, 15-Jul-2018.)
Assertion
Ref Expression
exanaliim (∃𝑥(𝜑 ∧ ¬ 𝜓) → ¬ ∀𝑥(𝜑𝜓))

Proof of Theorem exanaliim
StepHypRef Expression
1 annimim 676 . . 3 ((𝜑 ∧ ¬ 𝜓) → ¬ (𝜑𝜓))
21eximi 1580 . 2 (∃𝑥(𝜑 ∧ ¬ 𝜓) → ∃𝑥 ¬ (𝜑𝜓))
3 exnalim 1626 . 2 (∃𝑥 ¬ (𝜑𝜓) → ¬ ∀𝑥(𝜑𝜓))
42, 3syl 14 1 (∃𝑥(𝜑 ∧ ¬ 𝜓) → ¬ ∀𝑥(𝜑𝜓))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103  ∀wal 1330  ∃wex 1469 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-17 1507  ax-ial 1515 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-fal 1338  df-nf 1438 This theorem is referenced by:  rexnalim  2428  nssr  3163  nssssr  4154  brprcneu  5425
 Copyright terms: Public domain W3C validator