Theorem bj-nnst 13624

Description: Double negation of stability of a formula. Intuitionistic logic refutes unstability (but does not prove stability) of any formula. This theorem can also be proved in classical refutability calculus (see https://us.metamath.org/mpeuni/bj-peircestab.html) but not in minimal calculus (see https://us.metamath.org/mpeuni/bj-stabpeirce.html). See nnnotnotr 13872 for the version not using the definition of stability. (Contributed by BJ, 9-Oct-2019.) Prove it in ( → , ¬ ) -intuitionistic calculus with definitions (uses of ax-ia1 105, ax-ia2 106, ax-ia3 107 are via sylibr 133, necessary for definition unpackaging), and in ( → , ↔ , ¬ )-intuitionistic calculus, following a discussion with Jim Kingdon. (Revised by BJ, 27-Oct-2024.)

Assertion

Ref	Expression
bj-nnst	⊢ ¬ ¬ STAB 𝜑

Proof of Theorem bj-nnst

Step	Hyp	Ref	Expression
1		bj-trst 13620	. 2 ⊢ (𝜑 → STAB 𝜑)
2		bj-fast 13622	. 2 ⊢ (¬ 𝜑 → STAB 𝜑)
3	1, 2	bj-imnimnn 13619	1 ⊢ ¬ ¬ STAB 𝜑

Syntax hints:  ¬ wn 3  STAB wstab 820

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

This theorem depends on definitions:  df-bi 116  df-stab 821

This theorem is referenced by:  bj-stst  13626  bj-dcst  13642