Theorem bj-stst 12978

Description: Stability of a proposition is stable if and only if that proposition is stable. STAB is idempotent. (Contributed by BJ, 9-Oct-2019.)

Assertion

Ref	Expression
bj-stst	⊢ (STAB STAB 𝜑 ↔ STAB 𝜑)

Proof of Theorem bj-stst

Step	Hyp	Ref	Expression
1		bj-nnst 12974	. 2 ⊢ ¬ ¬ STAB 𝜑
2		bj-nnbist 12963	. 2 ⊢ (¬ ¬ STAB 𝜑 → (STAB STAB 𝜑 ↔ STAB 𝜑))
3	1, 2	ax-mp 5	1 ⊢ (STAB STAB 𝜑 ↔ STAB 𝜑)

Syntax hints:  ¬ wn 3   ↔ wb 104  STAB wstab 815

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 603  ax-in2 604  ax-io 698

This theorem depends on definitions:  df-bi 116  df-stab 816  df-dc 820

This theorem is referenced by: (None)