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

Theorem nnral 2465
Description: The double negation of a universal quantification implies the universal quantification of the double negation. Restricted quantifier version of nnal 1647. (Contributed by Jim Kingdon, 1-Aug-2024.)
Assertion
Ref Expression
nnral (¬ ¬ ∀𝑥𝐴 𝜑 → ∀𝑥𝐴 ¬ ¬ 𝜑)

Proof of Theorem nnral
StepHypRef Expression
1 rexnalim 2464 . . 3 (∃𝑥𝐴 ¬ 𝜑 → ¬ ∀𝑥𝐴 𝜑)
21con3i 632 . 2 (¬ ¬ ∀𝑥𝐴 𝜑 → ¬ ∃𝑥𝐴 ¬ 𝜑)
3 ralnex 2463 . 2 (∀𝑥𝐴 ¬ ¬ 𝜑 ↔ ¬ ∃𝑥𝐴 ¬ 𝜑)
42, 3sylibr 134 1 (¬ ¬ ∀𝑥𝐴 𝜑 → ∀𝑥𝐴 ¬ ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wral 2453  wrex 2454
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 1445  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-4 1508  ax-17 1524  ax-ial 1532
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-nf 1459  df-ral 2458  df-rex 2459
This theorem is referenced by:  onntri13  7227  onntri24  7231
  Copyright terms: Public domain W3C validator