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Mirrors > Home > ILE Home > Th. List > dcstab | GIF version |
Description: Decidability implies stability. The converse need not hold. (Contributed by David A. Wheeler, 13-Aug-2018.) |
Ref | Expression |
---|---|
dcstab | ⊢ (DECID 𝜑 → STAB 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | notnotrdc 838 | . 2 ⊢ (DECID 𝜑 → (¬ ¬ 𝜑 → 𝜑)) | |
2 | df-stab 826 | . 2 ⊢ (STAB 𝜑 ↔ (¬ ¬ 𝜑 → 𝜑)) | |
3 | 1, 2 | sylibr 133 | 1 ⊢ (DECID 𝜑 → STAB 𝜑) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 STAB wstab 825 DECID wdc 829 |
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-in2 610 ax-io 704 |
This theorem depends on definitions: df-bi 116 df-stab 826 df-dc 830 |
This theorem is referenced by: stdcndc 840 stdcndcOLD 841 condc 848 imandc 884 sbthlemi3 6932 |
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