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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-stst | GIF version |
Description: Stability of a proposition is stable if and only if that proposition is stable. STAB is idempotent. (Contributed by BJ, 9-Oct-2019.) |
Ref | Expression |
---|---|
bj-stst | ⊢ (STAB STAB 𝜑 ↔ STAB 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-nnst 13743 | . 2 ⊢ ¬ ¬ STAB 𝜑 | |
2 | bj-nnbist 13744 | . 2 ⊢ (¬ ¬ STAB 𝜑 → (STAB STAB 𝜑 ↔ STAB 𝜑)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (STAB STAB 𝜑 ↔ STAB 𝜑) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ↔ wb 104 STAB wstab 825 |
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 609 ax-in2 610 |
This theorem depends on definitions: df-bi 116 df-stab 826 |
This theorem is referenced by: (None) |
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