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Theorem anordc 874
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, pm3.1 681, 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 853 . 2 (DECID 𝜑 → (DECID 𝜓DECID (𝜑𝜓)))
2 ianordc 810 . . . . 5 (DECID 𝜑 → (¬ (𝜑𝜓) ↔ (¬ 𝜑 ∨ ¬ 𝜓)))
32bicomd 133 . . . 4 (DECID 𝜑 → ((¬ 𝜑 ∨ ¬ 𝜓) ↔ ¬ (𝜑𝜓)))
43a1d 22 . . 3 (DECID 𝜑 → (DECID (𝜑𝜓) → ((¬ 𝜑 ∨ ¬ 𝜓) ↔ ¬ (𝜑𝜓))))
54con2biddc 785 . 2 (DECID 𝜑 → (DECID (𝜑𝜓) → ((𝜑𝜓) ↔ ¬ (¬ 𝜑 ∨ ¬ 𝜓))))
61, 5syld 44 1 (DECID 𝜑 → (DECID 𝜓 → ((𝜑𝜓) ↔ ¬ (¬ 𝜑 ∨ ¬ 𝜓))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 101  wb 102  wo 639  DECID wdc 753
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640
This theorem depends on definitions:  df-bi 114  df-dc 754
This theorem is referenced by:  pm3.11dc  875  dn1dc  878
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