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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-dcstab | GIF version |
Description: A decidable formula is stable. (Contributed by BJ, 24-Nov-2023.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-dcstab | ⊢ (DECID 𝜑 → STAB 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dc 835 | . 2 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑)) | |
2 | bj-trst 14113 | . . 3 ⊢ (𝜑 → STAB 𝜑) | |
3 | bj-fast 14115 | . . 3 ⊢ (¬ 𝜑 → STAB 𝜑) | |
4 | 2, 3 | jaoi 716 | . 2 ⊢ ((𝜑 ∨ ¬ 𝜑) → STAB 𝜑) |
5 | 1, 4 | sylbi 121 | 1 ⊢ (DECID 𝜑 → STAB 𝜑) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 708 STAB wstab 830 DECID wdc 834 |
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-in2 615 ax-io 709 |
This theorem depends on definitions: df-bi 117 df-stab 831 df-dc 835 |
This theorem is referenced by: bj-nnbidc 14131 |
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