Theorem bj-nnst 14055

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 14302 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 106, ax-ia2 107, ax-ia3 108 are via sylibr 134, 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 14051	. 2 ⊢ (𝜑 → STAB 𝜑)
2		bj-fast 14053	. 2 ⊢ (¬ 𝜑 → STAB 𝜑)
3	1, 2	bj-imnimnn 14050	1 ⊢ ¬ ¬ STAB 𝜑

Syntax hints:  ¬ wn 3  STAB wstab 830

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

This theorem depends on definitions:  df-bi 117  df-stab 831

This theorem is referenced by:  bj-stst  14057  bj-dcst  14073