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Mirrors > Home > ILE Home > Th. List > anordc | GIF version |
Description: Conjunction in terms of disjunction (DeMorgan's law). Theorem *4.5 of [WhiteheadRussell] p. 120, but where the propositions are decidable. The forward direction, pm3.1 755, holds for all propositions, but the equivalence only holds given decidability. (Contributed by Jim Kingdon, 21-Apr-2018.) |
Ref | Expression |
---|---|
anordc | ⊢ (DECID 𝜑 → (DECID 𝜓 → ((𝜑 ∧ 𝜓) ↔ ¬ (¬ 𝜑 ∨ ¬ 𝜓)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dcan 935 | . . 3 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → DECID (𝜑 ∧ 𝜓)) | |
2 | 1 | ex 115 | . 2 ⊢ (DECID 𝜑 → (DECID 𝜓 → DECID (𝜑 ∧ 𝜓))) |
3 | ianordc 900 | . . . . 5 ⊢ (DECID 𝜑 → (¬ (𝜑 ∧ 𝜓) ↔ (¬ 𝜑 ∨ ¬ 𝜓))) | |
4 | 3, 3, 3 | 3bitr2rd 217 | . . . 4 ⊢ (DECID 𝜑 → ((¬ 𝜑 ∨ ¬ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓))) |
5 | 4 | a1d 22 | . . 3 ⊢ (DECID 𝜑 → (DECID (𝜑 ∧ 𝜓) → ((¬ 𝜑 ∨ ¬ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓)))) |
6 | 5 | con2biddc 881 | . 2 ⊢ (DECID 𝜑 → (DECID (𝜑 ∧ 𝜓) → ((𝜑 ∧ 𝜓) ↔ ¬ (¬ 𝜑 ∨ ¬ 𝜓)))) |
7 | 2, 6 | syld 45 | 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 ∨ wo 709 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: pm3.11dc 959 dn1dc 962 |
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