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

Theorem exalim 1432
Description: One direction of a classical definition of existential quantification. One direction of Definition of [Margaris] p. 49. For a decidable proposition, this is an equivalence, as seen as dfexdc 1431. (Contributed by Jim Kingdon, 29-Jul-2018.)
Assertion
Ref Expression
exalim (∃𝑥𝜑 → ¬ ∀𝑥 ¬ 𝜑)

Proof of Theorem exalim
StepHypRef Expression
1 alnex 1429 . . 3 (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
21biimpi 118 . 2 (∀𝑥 ¬ 𝜑 → ¬ ∃𝑥𝜑)
32con2i 590 1 (∃𝑥𝜑 → ¬ ∀𝑥 ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wal 1283  wex 1422
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
This theorem is referenced by:  n0rf  3278  ax9vsep  3927
  Copyright terms: Public domain W3C validator