Theorem dfnan2 1494

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 1492	. 2 ⊢ ((𝜑 ⊼ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓))
2		imnan 399	. 2 ⊢ ((𝜑 → ¬ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓))
3	1, 2	bitr4i 278	1 ⊢ ((𝜑 ⊼ 𝜓) ↔ (𝜑 → ¬ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ⊼ wnan 1491

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 207  df-an 396  df-nan 1492

This theorem is referenced by:  nancom  1496  nannan  1497  nannot  1499  nanbi1  1501