Theorem ianordc 798

Description: Negated conjunction in terms of disjunction (DeMorgan's law). Theorem *4.51 of [WhiteheadRussell] p. 120, but where one proposition is decidable. The reverse direction, pm3.14 669, holds for all propositions, but the equivalence only holds where one proposition is decidable. (Contributed by Jim Kingdon, 21-Apr-2018.)

Assertion

Ref	Expression
ianordc	⊢ (DECID φ → (¬ (φ ∧ ψ) ↔ (¬ φ ∨ ¬ ψ)))

Proof of Theorem ianordc

Step	Hyp	Ref	Expression
1		imnan 623	. 2 ⊢ ((φ → ¬ ψ) ↔ ¬ (φ ∧ ψ))
2		pm4.62dc 797	. 2 ⊢ (DECID φ → ((φ → ¬ ψ) ↔ (¬ φ ∨ ¬ ψ)))
3	1, 2	syl5bbr 183	1 ⊢ (DECID φ → (¬ (φ ∧ ψ) ↔ (¬ φ ∨ ¬ ψ)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 628  DECID wdc 741

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 629

This theorem depends on definitions:  df-bi 110  df-dc 742

This theorem is referenced by:  anordc  862  19.33bdc  1518  nn0n0n1ge2b  8096