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Mirrors > Home > ILE Home > Th. List > stdcn | GIF version |
Description: A formula is stable if and only if the decidability of its negation implies its decidability. Note that the right-hand side of this biconditional is the converse of dcn 837. (Contributed by BJ, 18-Nov-2023.) |
Ref | Expression |
---|---|
stdcn | ⊢ (STAB 𝜑 ↔ (DECID ¬ 𝜑 → DECID 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | stdcndc 840 | . . . 4 ⊢ ((STAB 𝜑 ∧ DECID ¬ 𝜑) ↔ DECID 𝜑) | |
2 | 1 | biimpi 119 | . . 3 ⊢ ((STAB 𝜑 ∧ DECID ¬ 𝜑) → DECID 𝜑) |
3 | 2 | ex 114 | . 2 ⊢ (STAB 𝜑 → (DECID ¬ 𝜑 → DECID 𝜑)) |
4 | olc 706 | . . . . 5 ⊢ (¬ ¬ 𝜑 → (¬ 𝜑 ∨ ¬ ¬ 𝜑)) | |
5 | 4 | imim1i 60 | . . . 4 ⊢ (((¬ 𝜑 ∨ ¬ ¬ 𝜑) → (𝜑 ∨ ¬ 𝜑)) → (¬ ¬ 𝜑 → (𝜑 ∨ ¬ 𝜑))) |
6 | orel2 721 | . . . 4 ⊢ (¬ ¬ 𝜑 → ((𝜑 ∨ ¬ 𝜑) → 𝜑)) | |
7 | 5, 6 | sylcom 28 | . . 3 ⊢ (((¬ 𝜑 ∨ ¬ ¬ 𝜑) → (𝜑 ∨ ¬ 𝜑)) → (¬ ¬ 𝜑 → 𝜑)) |
8 | df-dc 830 | . . . 4 ⊢ (DECID ¬ 𝜑 ↔ (¬ 𝜑 ∨ ¬ ¬ 𝜑)) | |
9 | df-dc 830 | . . . 4 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑)) | |
10 | 8, 9 | imbi12i 238 | . . 3 ⊢ ((DECID ¬ 𝜑 → DECID 𝜑) ↔ ((¬ 𝜑 ∨ ¬ ¬ 𝜑) → (𝜑 ∨ ¬ 𝜑))) |
11 | df-stab 826 | . . 3 ⊢ (STAB 𝜑 ↔ (¬ ¬ 𝜑 → 𝜑)) | |
12 | 7, 10, 11 | 3imtr4i 200 | . 2 ⊢ ((DECID ¬ 𝜑 → DECID 𝜑) → STAB 𝜑) |
13 | 3, 12 | impbii 125 | 1 ⊢ (STAB 𝜑 ↔ (DECID ¬ 𝜑 → DECID 𝜑)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∨ wo 703 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-in1 609 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: dcnn 843 bj-charfunbi 13806 |
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