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Mirrors > Home > ILE Home > Th. List > stdcndcOLD | GIF version |
Description: Obsolete version of stdcndc 845 as of 28-Oct-2023. (Contributed by David A. Wheeler, 13-Aug-2018.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
stdcndcOLD | ⊢ ((STAB 𝜑 ∧ DECID ¬ 𝜑) ↔ DECID 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exmiddc 836 | . . . . . 6 ⊢ (DECID ¬ 𝜑 → (¬ 𝜑 ∨ ¬ ¬ 𝜑)) | |
2 | 1 | adantl 277 | . . . . 5 ⊢ ((STAB 𝜑 ∧ DECID ¬ 𝜑) → (¬ 𝜑 ∨ ¬ ¬ 𝜑)) |
3 | df-stab 831 | . . . . . . . 8 ⊢ (STAB 𝜑 ↔ (¬ ¬ 𝜑 → 𝜑)) | |
4 | 3 | biimpi 120 | . . . . . . 7 ⊢ (STAB 𝜑 → (¬ ¬ 𝜑 → 𝜑)) |
5 | 4 | orim2d 788 | . . . . . 6 ⊢ (STAB 𝜑 → ((¬ 𝜑 ∨ ¬ ¬ 𝜑) → (¬ 𝜑 ∨ 𝜑))) |
6 | 5 | adantr 276 | . . . . 5 ⊢ ((STAB 𝜑 ∧ DECID ¬ 𝜑) → ((¬ 𝜑 ∨ ¬ ¬ 𝜑) → (¬ 𝜑 ∨ 𝜑))) |
7 | 2, 6 | mpd 13 | . . . 4 ⊢ ((STAB 𝜑 ∧ DECID ¬ 𝜑) → (¬ 𝜑 ∨ 𝜑)) |
8 | 7 | orcomd 729 | . . 3 ⊢ ((STAB 𝜑 ∧ DECID ¬ 𝜑) → (𝜑 ∨ ¬ 𝜑)) |
9 | df-dc 835 | . . 3 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑)) | |
10 | 8, 9 | sylibr 134 | . 2 ⊢ ((STAB 𝜑 ∧ DECID ¬ 𝜑) → DECID 𝜑) |
11 | dcstab 844 | . . 3 ⊢ (DECID 𝜑 → STAB 𝜑) | |
12 | dcn 842 | . . 3 ⊢ (DECID 𝜑 → DECID ¬ 𝜑) | |
13 | 11, 12 | jca 306 | . 2 ⊢ (DECID 𝜑 → (STAB 𝜑 ∧ DECID ¬ 𝜑)) |
14 | 10, 13 | impbii 126 | 1 ⊢ ((STAB 𝜑 ∧ DECID ¬ 𝜑) ↔ DECID 𝜑) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ 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-in1 614 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: (None) |
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