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Mirrors > Home > ILE Home > Th. List > annimdc | GIF version |
Description: Express conjunction in terms of implication. The forward direction, annimim 687, is valid for all propositions, but as an equivalence, it requires a decidability condition. (Contributed by Jim Kingdon, 25-Apr-2018.) |
Ref | Expression |
---|---|
annimdc | ⊢ (DECID 𝜑 → (DECID 𝜓 → ((𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑 → 𝜓)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imandc 890 | . . . 4 ⊢ (DECID 𝜓 → ((𝜑 → 𝜓) ↔ ¬ (𝜑 ∧ ¬ 𝜓))) | |
2 | 1 | adantl 277 | . . 3 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → ((𝜑 → 𝜓) ↔ ¬ (𝜑 ∧ ¬ 𝜓))) |
3 | dcim 842 | . . . . 5 ⊢ (DECID 𝜑 → (DECID 𝜓 → DECID (𝜑 → 𝜓))) | |
4 | 3 | imp 124 | . . . 4 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → DECID (𝜑 → 𝜓)) |
5 | dcn 843 | . . . . 5 ⊢ (DECID 𝜓 → DECID ¬ 𝜓) | |
6 | dcan 935 | . . . . 5 ⊢ ((DECID 𝜑 ∧ DECID ¬ 𝜓) → DECID (𝜑 ∧ ¬ 𝜓)) | |
7 | 5, 6 | sylan2 286 | . . . 4 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → DECID (𝜑 ∧ ¬ 𝜓)) |
8 | con2bidc 876 | . . . 4 ⊢ (DECID (𝜑 → 𝜓) → (DECID (𝜑 ∧ ¬ 𝜓) → (((𝜑 → 𝜓) ↔ ¬ (𝜑 ∧ ¬ 𝜓)) ↔ ((𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑 → 𝜓))))) | |
9 | 4, 7, 8 | sylc 62 | . . 3 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → (((𝜑 → 𝜓) ↔ ¬ (𝜑 ∧ ¬ 𝜓)) ↔ ((𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑 → 𝜓)))) |
10 | 2, 9 | mpbid 147 | . 2 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → ((𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑 → 𝜓))) |
11 | 10 | ex 115 | 1 ⊢ (DECID 𝜑 → (DECID 𝜓 → ((𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑 → 𝜓)))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 DECID wdc 835 |
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 615 ax-in2 616 ax-io 710 |
This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 |
This theorem is referenced by: xordidc 1410 |
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