Theorem dfnan2 1514

Description: Alternative denial in terms of our primitive connectives (implication and negation). (Contributed by WL, 26-Jun-2020.)

Assertion

Ref	Expression
dfnan2	⊢ ((𝜑 ⊼ 𝜓) ↔ (𝜑 → ¬ 𝜓))

Proof of Theorem dfnan2

Step	Hyp	Ref	Expression
1		df-nan 1512	. 2 ⊢ ((𝜑 ⊼ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓))
2		imnan 403	. 2 ⊢ ((𝜑 → ¬ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓))
3	1, 2	bitr4i 280	1 ⊢ ((𝜑 ⊼ 𝜓) ↔ (𝜑 → ¬ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   ⊼ wnan 1511

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 209  df-an 400  df-nan 1512

This theorem is referenced by:  nancom  1516  nannan  1517  nannot  1519  nanbi1  1521