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Theorem anordc 958
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.)
Assertion
Ref Expression
anordc (DECID 𝜑 → (DECID 𝜓 → ((𝜑𝜓) ↔ ¬ (¬ 𝜑 ∨ ¬ 𝜓))))

Proof of Theorem anordc
StepHypRef Expression
1 dcan 935 . . 3 ((DECID 𝜑DECID 𝜓) → DECID (𝜑𝜓))
21ex 115 . 2 (DECID 𝜑 → (DECID 𝜓DECID (𝜑𝜓)))
3 ianordc 900 . . . . 5 (DECID 𝜑 → (¬ (𝜑𝜓) ↔ (¬ 𝜑 ∨ ¬ 𝜓)))
43, 3, 33bitr2rd 217 . . . 4 (DECID 𝜑 → ((¬ 𝜑 ∨ ¬ 𝜓) ↔ ¬ (𝜑𝜓)))
54a1d 22 . . 3 (DECID 𝜑 → (DECID (𝜑𝜓) → ((¬ 𝜑 ∨ ¬ 𝜓) ↔ ¬ (𝜑𝜓))))
65con2biddc 881 . 2 (DECID 𝜑 → (DECID (𝜑𝜓) → ((𝜑𝜓) ↔ ¬ (¬ 𝜑 ∨ ¬ 𝜓))))
72, 6syld 45 1 (DECID 𝜑 → (DECID 𝜓 → ((𝜑𝜓) ↔ ¬ (¬ 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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