Theorem anordc 897

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, pm3.1 703, 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 875	. 2 ⊢ (DECID 𝜑 → (DECID 𝜓 → DECID (𝜑 ∧ 𝜓)))
2		ianordc 832	. . . . 5 ⊢ (DECID 𝜑 → (¬ (𝜑 ∧ 𝜓) ↔ (¬ 𝜑 ∨ ¬ 𝜓)))
3	2	bicomd 139	. . . 4 ⊢ (DECID 𝜑 → ((¬ 𝜑 ∨ ¬ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓)))
4	3	a1d 22	. . 3 ⊢ (DECID 𝜑 → (DECID (𝜑 ∧ 𝜓) → ((¬ 𝜑 ∨ ¬ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓))))
5	4	con2biddc 807	. 2 ⊢ (DECID 𝜑 → (DECID (𝜑 ∧ 𝜓) → ((𝜑 ∧ 𝜓) ↔ ¬ (¬ 𝜑 ∨ ¬ 𝜓))))
6	1, 5	syld 44	1 ⊢ (DECID 𝜑 → (DECID 𝜓 → ((𝜑 ∧ 𝜓) ↔ ¬ (¬ 𝜑 ∨ ¬ 𝜓))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102   ↔ wb 103   ∨ wo 661  DECID wdc 775

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662

This theorem depends on definitions:  df-bi 115  df-dc 776

This theorem is referenced by:  pm3.11dc  898  dn1dc  901