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 → ((𝜑 ∧ 𝜓) ↔ ¬ (¬ 𝜑 ∨ ¬ 𝜓))))

Proof of Theorem anordc

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))))

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