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Theorem anordc 863
 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, pm3.1 671, holds for all propositions, but the equivalence only holds given decidability. (Contributed by Jim Kingdon, 21-Apr-2018.)
Assertion
Ref Expression
anordc (DECID 𝜑 → (DECID 𝜓 → ((𝜑𝜓) ↔ ¬ (¬ 𝜑 ∨ ¬ 𝜓))))

Proof of Theorem anordc
StepHypRef Expression
1 dcan 842 . 2 (DECID 𝜑 → (DECID 𝜓DECID (𝜑𝜓)))
2 ianordc 799 . . . . 5 (DECID 𝜑 → (¬ (𝜑𝜓) ↔ (¬ 𝜑 ∨ ¬ 𝜓)))
32bicomd 129 . . . 4 (DECID 𝜑 → ((¬ 𝜑 ∨ ¬ 𝜓) ↔ ¬ (𝜑𝜓)))
43a1d 22 . . 3 (DECID 𝜑 → (DECID (𝜑𝜓) → ((¬ 𝜑 ∨ ¬ 𝜓) ↔ ¬ (𝜑𝜓))))
54con2biddc 774 . 2 (DECID 𝜑 → (DECID (𝜑𝜓) → ((𝜑𝜓) ↔ ¬ (¬ 𝜑 ∨ ¬ 𝜓))))
61, 5syld 40 1 (DECID 𝜑 → (DECID 𝜓 → ((𝜑𝜓) ↔ ¬ (¬ 𝜑 ∨ ¬ 𝜓))))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 629  DECID wdc 742 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630 This theorem depends on definitions:  df-bi 110  df-dc 743 This theorem is referenced by:  pm3.11dc  864  dn1dc  867
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